MARGIN CALL RISK MANAGEMENT WITH FUTURES AND OPTIONS

A THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OFMIDDLE EAST TECHNICAL UNIVERSITY

BY

MURAT ALIRAVCI

IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR

THE DEGREE OF MASTER OF SCIENCEIN

INDUSTRIAL ENGINEERING

JANUARY 2013

Approval of the thesis:

MARGIN CALL RISK MANAGEMENT WITH FUTURES AND OPTIONS

submitted by MURAT ALIRAVCI in partial fulfillment of the requirements for the degree of Masterof Science in Industrial Engineering Department, Middle East Technical University by,

Prof. Dr. Canan ÖzgenDean, Graduate School of Natural and Applied Sciences

Prof. Dr. Sinan KayalıgilHead of Department, Industrial Engineering

Assist. Prof. Dr. Serhan DuranSupervisor, Industrial Engineering Dept., METU

Assist. Prof. Dr. Fehmi TanrıseverCo-supervisor, Industrial Engineering and Innovation Sciences Dept., TU/e

Examining Committee Members:

Assist. Prof. Dr. Ismail Serdar BakalIndustrial Engineering Dept., METU

Assist. Prof. Dr. Serhan DuranIndustrial Engineering Dept., METU

Assist. Prof. Dr. Fehmi TanrıseverIndustrial Engineering and Innovation Sciences Dept., TU/e

Assoc. Prof. Dr. Zeynep Pelin BayındırIndustrial Engineering Dept., METU

Assoc. Prof. Dr. Sinan GürelIndustrial Engineering Dept., METU

Date:

I hereby declare that all information in this document has been obtained and presented in ac-cordance with academic rules and ethical conduct. I also declare that, as required by these rulesand conduct, I have fully cited and referenced all material and results that are not original tothis work.

Name, Last Name: MURAT ALIRAVCI

Signature :

iv

ABSTRACT

MARGIN CALL RISK MANAGEMENT WITH FUTURES AND OPTIONS

Alıravcı, MuratM.S., Department of Industrial EngineeringSupervisor : Assist. Prof. Dr. Serhan DuranCo-Supervisor : Assist. Prof. Dr. Fehmi Tanrısever

January 2013, 85 pages

This study examines dynamic hedge policy of a company in a multi-period framework. The companybegins to operate a project for a customer and it also has a subcontractor which completes an importantpart of the project by using an economic commodity. The customer will pay a fixed price to thecompany at the end of the project. Meanwhile, the company needs to pay the debt to the subcontractorand the amount of the debt depends on the spot price of the commodity at that time. The companyis allowed to hedge for the commodity price fluctuations via future and option contracts. Since thecompany has a limited cash reserve as well as previously planned payments, it may face financialdistress when the net cash balance decreases below zero. Consequently, the company maximizes theexpected value of itself by minimizing the expected financial distress cost.

Keywords: Risk Management, Price Risk, Margin Call, Futures, Options

v

ÖZ

VADELI ISLEMLER VE OPSIYONLAR ILE TEMINAT TAMAMLAMA ÇAGRI RISKYÖNETIMI

Alıravcı, MuratYüksek Lisans, Endüstri Mühendisligi BölümüTez Yöneticisi : Yrd. Doç. Dr. Serhan DuranOrtak Tez Yöneticisi : Yrd. Doç. Dr. Fehmi Tanrısever

Ocak 2013, 85 sayfa

Bu çalısma bir sirketin dinamik olarak riske karsı korunma politikasını çok periyotlu bir çerçevedeincelemektedir. Sirketin müsterisi için bir proje yapım sorumlulugu bulunmakta ve belirli bir emtiakullanarak projenin önemli bir bölümünü gerçeklestirecek bir alt yükleniciye bu projede ihtiyaç duy-maktadır. Müsteri bu proje baslangıcında belirlenen bir fiyatı proje sonunda sirkete ödeyecektir. Yineproje sonunda, alt yükleniciye borcunu ödeyecek ve borcun miktarı emtianın o andaki piyasa fiyatıyladogrudan baglantılı olacaktır. Bu senaryo kapsamında sirket, emtiaya baglı vadeli türev islemleri veopsiyonlar aracılıgıyla riskten korunabilir. Sirketin nakit hesap dengesinin sınırlılıgı ve daha önce-den planlanmıs ödemelerinin varlıgı sebebiyle nakit hesabının sıfırın altına indigi durumlarda sirkettemali sıkıntılar yasanabilir. Bu sebeple sirket, gelecekteki beklenen degerini en üst düzeye çıkarmayaçalısırken nakit yetmezliginden kaynaklanan mali sorunların beklenen maliyetini en düsük düzeye in-dirmelidir.

Anahtar Kelimeler: Risk Yönetimi, Fiyat Riski, Teminat Tamamlama Çagrısı, Vadeli Islemler, Opsiy-

onlar

vi

ACKNOWLEDGMENTS

I would like to use this opportunity to express my thanks to all people who has supported me duringmy studies.

First of all I would like to thank to my advisors, Serhan Duran and Fehmi Tanrısever, for their greatguidance during my study. Without their contribution both in academical and personal life, I wouldnot be able to successfully finish this study. They are academically precious guides as well as beinggreat friends to me.

I would like to thank to my instructors Pelin Bayındır and Sedef Meral for supporting me to enrol intothis academical program.

I also would like to thank to my all friends for their supports during this process. Özgün Özkan,Mustafa Onur Özdemir, Burakhan Kocaman, Semih Özdede, Ilker Özbakır and many others madethe life in Ankara more enjoyable. Besides, my friends in Eindhoven helped me to gain a uniqueexperience there.

Finally, I would like to thank my family for their huge supports in every field they are able to contribute.They have always been there when needed and nothing good would possibly happen in my life withoutthem.

vii

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

CHAPTERS

1 INTRODUCTION AND MOTIVATION . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 INTRODUCTION TO FINANCIAL DERIVATIVES . . . . . . . . . . . . . . . . . . 3

3 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 MATHEMATICAL MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1 One-Period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Two-Period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 Three-Period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.4 Approximation of Two-Period Model with Compound Interest . . . . . . . . 22

4.5 One-Period Model with Call Option . . . . . . . . . . . . . . . . . . . . . . . 23

5 NUMERICAL ANALYSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Analysis of Two-Period Model . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1.1 Analysis of Two-Period Model With Changing Values of 4 . . . . . 31

5.1.1.1 Effect of n3 while 4 values change: . . . . . . . . . . . 34

5.1.1.2 Effect of δ while 4 values change . . . . . . . . . . . 35

5.1.1.3 Effect of ξ while 4 values change . . . . . . . . . . . 36

5.1.2 Analysis of Two-Period Model With Changing Values of n3 . . . . 37

5.2 Analysis of Three-Period Model . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.1 Analysis of Three-Period Model With Changing Values of 4 . . . . 39

5.2.1.1 Effect of n3 while 4 values change . . . . . . . . . . . 41

5.3 Analysis of the Approximation of the Model with Compound Interest . . . . . 42

5.4 Analysis of One-Period Model With Call Option . . . . . . . . . . . . . . . . 45

6 CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

viii

APPENDICES

A LIST OF NOTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

B PROOFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

B.1 Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

B.2 Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

B.3 Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

B.4 Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

B.5 Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

B.6 Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

B.7 Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

B.8 Theorem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B.9 Theorem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

B.10 Theorem 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B.11 Theorem 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

B.12 Theorem 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B.13 Theorem 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.14 Theorem 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.15 Theorem 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

B.16 Theorem 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

ix

LIST OF TABLES

TABLES

Table 2.1 Cash flows when realized price is $ 11 in case of full hedge policy via forwards . . . 4

Table 2.2 Cash flows when realized price is $ 9 in case of full hedge policy via forwards . . . . 4

Table 2.3 Marking to market example for three days of operation . . . . . . . . . . . . . . . . 5

Table 2.4 Cash flows when realized price is $ 11 in case of full hedge policy via call options . . 5

Table 2.5 Cash flows when realized price is $ 9 in case of full hedge policy via call options . . 6

Table 4.1 Summary of events for one-period model . . . . . . . . . . . . . . . . . . . . . . . . 13

Table 4.2 Summary of the events for two-period model . . . . . . . . . . . . . . . . . . . . . . 16

Table 4.3 Summary of theorems for two-period model . . . . . . . . . . . . . . . . . . . . . . 19

Table 4.4 Summary of the events for three-period model . . . . . . . . . . . . . . . . . . . . . 20

Table 4.5 Summary of events for one-period model with call option . . . . . . . . . . . . . . . 25

x

LIST OF FIGURES

FIGURES

Figure 4.1 Summary of financial transactions for one-period model . . . . . . . . . . . . . . . 13

Figure 4.2 Summary of financial transactions for two-period model . . . . . . . . . . . . . . . 15

Figure 4.3 Summary of financial transactions for three-period model . . . . . . . . . . . . . . 20

Figure 4.4 Summary of financial transactions for one-period model with call option . . . . . . 24

Figure 5.1 Expected total financial distress costs for x1,3 values for two-period model with baseparameter setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Figure 5.2 Simulated optimum hedging decisions corresponding to changing values of 4 fortwo-period model with base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 5.3 Theoretical optimum hedging decisions corresponding for different values of 4 fortwo-period model with base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 5.4 Optimum hedging decisions corresponding to changing values of 4 for two-periodmodel with base parameters and different n3 levels . . . . . . . . . . . . . . . . . . . . . . 34

Figure 5.5 Optimum hedging decisions corresponding to changing values of 4 for two-periodmodel with base parameters and different δ levels . . . . . . . . . . . . . . . . . . . . . . 35

Figure 5.6 Optimum hedging decisions corresponding to changing values of 4 for two-periodmodel with base parameters and different ξ levels . . . . . . . . . . . . . . . . . . . . . . 36

Figure 5.7 Optimum hedging decisions corresponding to changing values of 4 for two-periodmodel with base parameters and different n3 levels . . . . . . . . . . . . . . . . . . . . . . 37

Figure 5.8 Simulated optimum hedging decisions corresponding to changing values of n3 fortwo-period model with base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Figure 5.9 Theoretical optimum hedging decisions corresponding to changing values of 4 fortwo-period model with base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Figure 5.10 Expected total financial distress costs for three-period model with base parametersfor different x1,4 values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Figure 5.11 Optimum hedging amounts at t = 1 corresponding to changing values of 4 forthree-period model with base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Figure 5.12 Optimum hedging decisions for t = 1 and expected optimum hedging decisions fort = 2 for three-period model at t = 1 with changing values of y1 and base parameters . . . 41

Figure 5.13 Optimum hedging decisions corresponding to changing values of 4 and n3 for three-period model with base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Figure 5.14 Optimum hedging decisions corresponding to changing values of 4 for two-periodmodel with compound interest and base parameters . . . . . . . . . . . . . . . . . . . . . 43

Figure 5.15 Optimum hedging decisions of two-period models with simple and compound in-terest corresponding to changing values of 4 by using base parameters (except β = 1) . . . 44

xi

Figure 5.16 Optimum hedging decisions of two-period models with simple and compound in-terest corresponding to changing values of 4 by using base parameters (except β = 1 andn3 = 70) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Figure 5.17 Optimum hedging decisions of two-period model with compound interest corre-sponding to changing values of 4 by using base parameters (except β = 1 and n3 = 50) . . . 45

Figure 5.18 Optimum hedging decisions of two-period model with compound interest corre-sponding to changing values of 4 and β by using base parameters . . . . . . . . . . . . . . 46

Figure 5.19 Expected total financial distress costs corresponding to different x1,2 values for one-period model with call options and with base parameters . . . . . . . . . . . . . . . . . . . 47

Figure 5.20 Optimum hedging decisions at t = 1 corresponding to changing values of 4 forone-period model with call options and with base parameters . . . . . . . . . . . . . . . . 47

Figure 5.21 Expected financial distress costs of one-period models with future and option con-tracts corresponding to different y1 values at optimum policy . . . . . . . . . . . . . . . . . 48

Figure B.1 Expected total financial distress costs by changing values of x1,3 for two-periodmodel with base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure B.2 Expected total financial distress costs by changing values of x1,3 for two-periodmodel with base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure B.3 Expected total financial distress costs by changing values of x1,3 for two-periodmodel with base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

xii

CHAPTER 1

INTRODUCTION AND MOTIVATION

In today’s economy, commodity prices in markets are important for companies that are utilizing themin their projects. When the prices are not predicted right and companies do not secure their financialpositions against risks by hedging, changes in prices may lead to decrease in revenue. Such unexpecteddecreases may lead distress in the financial situation of companies and financial distress costs mayincur when their cash balance becomes negative. The breakpoint values of the prices of commoditiesthat may lead financial distress are approximately calculated by considering the cash exchange of thecompany during different time periods of the project. In the time that companies reach an agreementwith their customers for the project that requires a specific delivery date in the future, companiesalready have planned deliverables or receivables to be realized during the project. Considering all thoseplanned deliverables or receivables as well as the initial cash balance, they are generating a financialplan during the project. However, if companies face a financial distress, a loan may be received fromthe bank with an interest rate higher than the minimum rate of return in the market. Financial distresscost may be accumulated by both simple and compound interest models. Accumulated distress costmay be paid in the beginning of the next period, or it may be delayed for the following periods. In orderto prevent the emergence of financial distress costs, hedging may be applied on the main commoditiesused during the project. However, the magnitude of the hedging decision and the derivatives used forit is a critical decision for companies.

In this paper, risk management decision of a firm which has an agreement to deliver a product mainlymanufactured by using a common commodity will be considered. The firm will receive a revenue atthe delivery of the products and will also pay the related debt to the subcontractor at that moment. Therevenue to be received from the customer is known at the beginning of the contract. However, the debtof the firm to the subcontractor is a variable that is affected by the current price of the commodity andthe profit margin of the subcontractor. Due to the price risk of the commodity, the firm may preferto use financial instruments; especially over-the-counter derivatives that are derived by the price ofthe commodity used by the firm. Indeed, buying or selling forward contracts, futures contracts andcall options maturing at the delivery time of the firm are probably the most common choice for suchcompanies.

In both of the interest rate models, financial distress cost is accumulated by multiplying the cash needof the firm with bank loan interest rate. However, both models behave differently when time horizonis more than a single period. If simple interest is used, the interest is calculated on original principalonly. However, compound interest takes original principal and all interest accumulated during pastperiods into account. Since the companies may be exposed to financial distress in some periods, someof the companies prefer to have overdraft arrangements with the banks. By using these agreements,the companies may borrow money from the bank and it will be exposed to a predefined interest ratefor its loan up to a predefined limit. Although both compound and simple interest rates may be used

for overdraft arrangements, simple interest rates are quite popular especially in case of long termcommitments between the bank and the companies. Simple interest rates may be higher in some cases,however it is more secure in case of an unexpected long term scarce of cash.

2

CHAPTER 2

INTRODUCTION TO FINANCIAL DERIVATIVES

Derivatives are financial instruments whose values are derived from one or more underlying variables.These variables may have a wide variety, such that commodities (steel, silver, oil, gas, etc.), finan-cial assets (bonds, stocks, etc.), another derivative (e.g. options on futures), index and interest rate.Forwards, futures and options will be mentioned in this paper.

Forwards are contracts that are held between two parties to buy or sell an asset at a predetermined priceand a predetermined delivery date in future. The contracts are traded in over-the-counter markets andare non-standardized, as a result the traders can exchange all kind of assets without limitation. Thus,it is a popular derivative in the market and mainly does not have initial payment. The positions of theparties in the contract are classified as long and short positions. Having a long position means agreeingto buy the asset. On the other hand, the party agreeing to sell an asset in the forward contract is takinga short position. Suppose that S i is the spot price of the asset at time i and F0,i is the agreed price bytwo parties at time 0 for the delivery time i. The payoff of the contract per unit is

(S i − F0,i

)for the

party with long position and it is(F0,i − S i

)for the party with short position at the end of the contract,

at time i. It means that taking long position lead to a profit in case that realized price of the commodityis higher than the future contract price, and it is vice versa for the short position. At the end, one partyhas a profit and this profit emerges as a cost to the other party.

Price risk may be completely eliminated by hedging through forwards. For instance, suppose that thereis a company which needs to buy 100 tons of a certain commodity at time 2 and tries to avoid the pricerisk by hedging for the whole amount at time 1. The forward price of the commodity for time 2 is $10 per ton at time 1 and there is no initial payment for the forward contract. In all cases, the firm willpay $ 1000 totally due to the hedging policy. Suppose that the realized spot price of the commoditybecomes $ 11 per ton. If the firm did not hedge at all at time 1, it would pay $ 1100 for the commoditypurchase at time 2 in this case. Therefore the firm has $ 1000 of profit due to hedging. On the otherhand, the firm would have a financial loss of $ 1000 if the spot price of the commodity would be $9 per ton, which is lower than the forward price. All financial transactions related to this scenario isshown at Table 2.1 and Table 2.2.

3

Table 2.1: Cash flows when realized price is $ 11 in case of full hedge policy via forwards

Table 2.2: Cash flows when realized price is $ 9 in case of full hedge policy via forwards

Futures are standardized contracts that are held between two parties to buy or sell specified asset withstandardized quantity and quality at a predetermined price and a predetermined delivery date in future,which are traded in over-the-counter markets. The party agreeing to buy the asset is said to be longpositioned, the party agreed to sell the asset is said to be short positioned. Both parties have to posta margin when futures contract is agreed. Suppose that S i is the spot price of the asset at time i, F0,i

is the agreed price by two parties at time 0 for the delivery time t and κ0 is the initial margin. Lastly,F j,i denotes the futures price of the commodity at time j for time i. When the change in the price ofthe future exceeds the margin, the party that has a negative payoff should post margin call. It meansthat the buyer needs to deposit a margin call κ j when the futures price at time j decreased more thanprepaid margin with respect to futures price at time 0, i.e. F0,i − F j,i = κ0. In the opposite case, i.e.F j,i − F0,i = κ0, the seller needs to deposit the margin. In this study, it will be assumed that there isno initial payment to enter into futures contract. However, the margin account of the firm needs to beupdated between the time periods.

Suppose that a firm buys 10 units of futures of a commodity at 1st of February for 1st at a price of $ 100per unit. It is also assumed that the initial margin for the contract is $ 50 and maintenance margin is $35; therefore the firm needs to pay $ 50 at the beginning of the contract and needs to compensate themargin account when it becomes less than $ 35. Daily margin account updates of the first three daysare shown in Table 2.3. In the case denoted in this table, the firm has a loss of $ 10 at 2nd of Februaryand balance of the margin account of the firm becomes $ 40 which is still more than maintenancemargin. Additionally, margin account becomess less than the maintenance margin at 3rd of Februaryand the firm needs to pay $ 40 to make its margin account $ 50 again. At 4th of February, the unitprice of the contract increases to $ 97 and the daily gain of the firm becomes $ 10.

4

Table 2.3: Marking to market example for three days of operation

Options are contracts that are held between two parties which gives to buyer (also called as optionholder) the right, but not the obligation, to buy or sell a specified asset at a predetermined quantity anda specified strike price on or before a specified date. The seller, in other words “option writer”, has tofulfil the transaction when the buyer chooses to exercise the option. The buyer pays a premium to theseller for this right at the beginning of the contract. The agreement is called “call” option when theagreement is based on buying an asset, and it is called “put” option when the agreement is based onselling. The position of buyer is called long position, while the position of seller is described as shortposition.

Suppose that there is a company which needs to buy 100 tons of a certain commodity at time 2 andtries to avoid the price risk by hedging via call options on futures for the whole amount at time 1. Thestrike price of the call option for time 2 is $ 10 per ton at time 1 and the company needs to pay a riskpremium of $ 0.5 per ton. Now, suppose that the realized spot price of the commodity becomes $ 11per ton. If the firm did not hedge at all at time 1, it would pay $ 1100 for the commodity purchase attime 2 in this case. However, the firm totally pays an amount of $ 50 at time 1 and $ 1000 at time 2due to hedging via call options, as shown in Table 2.4. On the other hand, the firm would totally pay $50 at time 1 and $ 900 at time 2 if the spot price of the commodity becomes $ 9 per ton of commodityas shown in Table 2.5.

Table 2.4: Cash flows when realized price is $ 11 in case of full hedge policy via call options

5

Table 2.5: Cash flows when realized price is $ 9 in case of full hedge policy via call options

6

CHAPTER 3

LITERATURE REVIEW

Companies try to secure their financial situation and avoid risks by hedging, especially in financiallycritical situations and low liquidity. They may use different kind of derivatives for hedging and mostcommon examples are forward, future and option contracts. Moreover, using financial derivatives isa growing practice in different industries nowadays. Therefore, there is a vast collection of litera-ture on using of derivatives and these studies are mainly divided mainly into three groups as utilitymaximization, value maximization and variance minimization problems.

The utility maximization models focus on utility function by considering risk approach of the deci-sionmakers. Indeed, most of the studies on this area focus on hedging of risk averse decisionmakers.The study of Heifner (1972) on finding optimal hedge policies for cattle feeding is an early example ofutility maximization models. Another example is Rolfo (1980) which focuses on optimal hedge policyfor cocoa producers with a risk averse approach. The paper considers both price and quantity risks withlogarithmic utility functions. On the other hand, variance minimization is another useful model usedin risk management problems. Minimizing variance reduces the unexpected consequences for the de-cisionmaker. The traditional method of static minimum-variance hedges are discussed by Stulz (2003)and McDonald (2006). One of the latest studies, Basak and Chabakauri (2012), provides a simple an-alytical solution for hedge decisions in a dynamic programming framework and discrete time settingsfor incomplete markets. Besides, Xing and Pietola (2005) figure out optimal hedge ratios of a wheatproducer under quantity and price risk using both mean-variance and expected utility frameworks.They suggest that the correlation between price and quantity is important in determination of the opti-mal hedge ratio. Their results suggest that price and quantity are negatively correlated which creates anatural hedge. They show that when there is natural hedge, the optimal hedge is always less than theexpected quantity. Moreover, they suggest that hedging effectiveness decreases as quantity uncertaintyincreases. These two methodologies are compared in a perspective of decisionmakers in some studies.For example, Cecchetti, Cumby, Figlewski (1988) suggest utility maximization approach rather thanvariance minimization on 20 year treasury bonds that would be held for one month periods.

Utility function maximization and variance minimization are properly applicable frameworks for in-dividual decisionmakers, risk averse and little sized companies, and may be more useful than otherapproaches. On the other hand, in the risky business environments of today’s economy, the companiesneed to manage the possible risks and seek for opportunities to grow. Therefore, variance minimiza-tion may not be an appropriate decision framework for many companies. Besides, utility maximizationmay not be applicable to the decision process of large corporations because there are a lot of stake-holders which have different types of risk approaches. Therefore, setting a common objective to beaccepted by all of the components is necessary.

Value maximization problems maximize the discounted values of the expected financial balance of the

7

company; as a result it is more prone to be used as the main decision criteria for large corporations.The study of Modigliani and Miller (1958) proved that hedging does not add any value to the value ofthe company in perfect markets. On the other hand, Smith and Stulz (1985) demonstrated that hedgingmay increase the expected value of the company in case of market imperfections such as managerialrisk aversion, financial distress costs, bankruptcy and taxes. Indeed, hedging is an important practice toprevent financial distress. A more recent study completed by data collecting from 400 UK companies,Judge (2006), shows that one of the most common reasons of hedging is avoiding financial distressand firms with higher cash balance are less likely to hedge.

In our study, value maximization approach is used to decrease the expected financial distress cost. Thecompany may face financial distress due to the price risk of the commodity used in its processes. Thereis a large literature about the effect of price risk on hedging decision; yet, a few of them use valuemaximization approach. One of them is the study of Maes (2011) which uses value maximizationapproach in order to analyze the hedging decisions of a wheat miller that is confronted with price,quantity and blend risks. It shows that when uncertainty increases, a lower optimal hedge ratio appears.Goel and Gutierrez (2009) and Goel and Gutierrez (2011) also use value maximization framework.

Goel and Gutierrez (2009) show the effects of changing futures and spot prices on inventory man-agement decision of a company. Likewise, Goel and Gutierrez (2011) examine the additional valuedue to procurement by using commodity markets. Additionally, Goel and Tanrisever (2011) analyzesthe hedge behavior of a company using a commodity in procurement, its processes and distribution.They analyze the optimum policy to balance procurement in spot market and forward/options mar-ket in a multi-period framework while the transportation cost paid for the spot market is higher thanforward/options market.

Tanrisever and Gutierrez (2011) examine the motives for hedge actions of a flour miller using valuemaximization under financial distress risk. Their study considers price risk of the commodity used andit shows that hedging increase the value of the firm. They showed that price risk can be eliminatedcompletely by hedging of the related commodities in make-to-order production and production levelmay be increased.

Margin call risk management is also another important characteristic of our study. Since margin callsmay create an unexpected cost to the company, literature partly focuses on avoiding margin calls due tofuture and option contracts. For example, Garner (2010) focuses on different hedge strategies in orderto reduce margin requirements of hedging via futures and options. Another characteristic approachis provided by the study of Dau and Groch (2005) which suggests a decision framework for buyingcontracts with margin call risk by using historical data to examine buying power of the company infuture for an automated risk management back office. One of the two studies that is largely benefitedin this paper is Tanrisever and Levij (2011) which examine the hedge policy of an electricity tradingcompany facing margin calls due to price risk in a value maximization perspective in a two-periodframework. Another beneficial study is Tanrisever, Duran and Sumer (2011) which focuses on hedgebehavior of an offshore wind-farm construction company which mainly uses steel for construction,under price and quantity risk in one and two-period framework. This study uses value maximizationapproach with compound interest rates and underhedging is proved to be optimal for two-period modelunder price risk.

Unlike the previous studies, we analyze the hedge policy of a company facing price risk by usingvalue maximization approach. One, two and multi-period analysis is completed by using futures andforward contracts, and the decision of buying call options in one-period of time is also analyzed inorder to benchmark with the usage of futures and forwards. Although simple interest rates are used by

8

assuming that there is an overdraft agreement between the company and a bank, an approximation forthe model with compound interest rates in two-period perspective is also examined.

9

10

CHAPTER 4

MATHEMATICAL MODELS

In this study, the hedging decision of a firm to manage its financial distress risk is examined. The firmundertakes an agreement with a customer to deliver a product at the agreed date. Customer will paya predefined price p per unit for a predefined quantity ξ at the agreement date. The firm outsources aconsiderable amount of work to a subcontractor which produces the necessary portion of the project.The debt to the subcontractor will also be paid at the end of the agreement when the firm gets paidby the customer. However, the main part of the cost paid to the subcontractor depends on the priceof the underlying commodity of the delivered asset at the end of the project. The remaining part isa fixed profit margin of the subcontractor. If the project is ending at time i, the total cost paid to thesubcontractor is (S i + λ) ξ for ξ units of assets where spot price of the asset at the end of the project isdenoted as S i and the profit margin of the subcontractor is denoted as λ.

Cash reserve of the company at time i after the transactions are realized (except any possible financialdistress) cost is denoted as yi. During the fulfillment process of the agreement, the firm will alsoreceive or pay its payments that are irrelevant to the project. These payments are already known atthe beginning of the project. A payment to be completed at time i is denoted as ni. The value of ni

may also be negative, in this case the company has receivables for time i. Additionally, the firm has anoverdraft agreement with a bank using simple interest rates. Therefore, in case of a negative balancein the cash balance of the firm, it may borrow money at a predefined simple interest rate. In this study,the firm’s account of loans from the bank will be totally evaluated after the project ends. For instance,the financial distress costs until time i (including financial distress cost at time i) will be taken into theaccount. The interest rate of bank loan is the same in all periods and denoted as r, as a result financialdistress cost due to scarce of money at the beginning of period i is denoted as r

[yi]−.

The aim of the company is to maximize the expected total change in the value of the company duringthe project. In the model, the firm is assumed to deal only with one project during the periods that areexamined, or the others are isolated from the decision. In order to maximize the total value change inpositive direction, the firm is able to long or short futures or call options. In case of futures are obtainedfor this aim, a higher hedging quantity from the beginning to the end of the project may decrease thefinancial distress risk at the delivery date of the project assets, yet it surely increases the financialdistress risk in the periods before the delivery date due to margin call of futures and it is vice versa forlower hedging quantity. On the other hand, options do not have margin calls and the risk premium ispaid at the beginning of the contract. Therefore, the firm pays the expected cost of the risk that willbe faced and does not have margin call updates during the contract is active. Although this approachmay prevent the financial distress due to unexpected price fluctuations, it decreases profitability due torisk premiums paid to purchase the contract and will be paid after the project ends (or the debt will beconsidered until the end of the project).

11

In this study, pricing of the futures contract of an asset is modeled as a stochastic process. IndeedF j+1,i has a normal distribution with mean F j,i and standard deviation σ. Finally the spot price ofthe underlying commodity S i has a mean of Fi−1,i and standard deviation σ. Therefore the followingexpression is valid for a decisionmaker at time j.

F j,i = Eφ(F j+1,i

)= Eφ

(F j+2,i

)= ... = Eφ

(Fi−1,i

)= Eφ

(S i

)However, for the same decisionmaker, the expected price of F j+2,i may not be equal to F j,i becausethe realized price of F j+1,i may be different than F j,i. Consequently, the following may be true for thedecisionmaker in time j + 1.

F j,i , F j+1,i = Eφ(F j+2,i

)= ... = Eφ

(Fi−1,i

)= Eφ

(S i

)In this study, the payoff of the call option will be examined with strike price of the future price of thecommodity maturing at the end of the delivery date of assets, i.e. F j,i. Thus, the risk premium of theoption will be calculated as h =

∫ ∞F1,2

(S 2 − F1,2

)φQ

S 2(S 2) dS 2 and this amount is needed to be paid by

the buyer when the call option is bought.

Finally, a minimum required rate of return is used to evaluate expected future transactions in theobjective function. The minimum required rate of return is the same for all periods and is denoted asrd. Besides β is defined in order to derive the value of a transaction of a discrete time to the previousone and it is formulized as 1/ (1 + rd).

Notation used in this table is provided briefly in Appendix.

4.1 One-Period Model

This case is assumed to be a scenario where there are no financial transactions between sign of thecontract and delivery of asset. The firm has a debt at the amount of n2 that is planned to be paid attime 2. It will receive the revenue from customer at an amount of pξ and will have to pay the cost ofthe assets to the subcontractor at an amount of

(S 2 + λ

)ξ at time 2. The cash balance of the firm at the

beginning of the project is y1.

In the beginning of the period, the firm considers to long x1,2 units of commodity futures contracts attime 1 maturing at time 2 for the futures price F1,2 while the spot price of the commodity is S 1 at thistime. By this action, the firm aims to decrease the risk of having financial distress cost at time 2 dueto unstable price of the commodity. In this part of the study, it is assumed that the account of the firmis only updated at time 2, similar to a forward contract there is no margin update during the period.Since initial margin that is to be modeled is not considered in the model, it is the same model withthe futures contract for one-period when margin call is not deposited during the project. The amountof cash transaction at time 2 led by hedging is denoted by

(S 2 − F1,2

)x1,2. It is also assumed that the

price of the commodity at time 2, which is denoted as S 2, has a normal distribution with mean F1,2 andstandard deviation σ. Another assumption is that the price of the futures contract maturing at a certaintime is equal to the expected spot price of the underlying commodity at that time. Thus the expectedvalue of the underlying commodity at time 2 is equal to the price of the futures contract maturing attime 2, i.e. F1,2 = Eφ

(S 2

). The decision variable for the firm is the amount of futures contract to buy

which is denoted as x1,2.

12

All financial transactions of the described problem are briefly shown at Figure 4.1.

Figure 4.1: Summary of financial transactions for one-period model

To sum up, the firm has only one available decision, it is setting of hedging quantity at the at time 1.Besides it has to borrow money with interest rate r if the cash balance becomes negative. All financialtransactions, decisions and actions of the company are shown together at the summary of events tablebelow, at Table 4.1.

Table 4.1: Summary of events for one-period model

In this study, it is assumed that the company aims to maximize the expected total value of the companyat the end of the contract. Therefore V j (yi) is used as a function decision variables which maximizesthe expected change in the value of the company at time j with an initial cash yi. Besides anotherfunction 4V j (yi) is defined to describe the change in the expected value of the company at the end

13

of period j with an initial cash yi. Finally, the objective function V j (yi) can be defined as a functionreferring the maximum value of the function 4V j (yi), which can be translated into a minimizationproblem by minimizing

(−4V j (yi)

). Another assumption is that the decisionmaker is risk neutral.

Since the current problem is solved in one period framework and the only decision variable is x1,2, theproblem of the firm can be formulated as follows:

Objective Function:

V2 (y1) = maxx1,24V2 (y1) = −min

x1,2(−4V2 (y1))

= minx1,2

Eφs2

[n2 −

(p − S 2 − λ

)ξ −

(S 2 − F1,2

)x1,2 + βr

[y2

]−]

Subject to:y2 = y1 − n2 +

(p − S 2 − λ

)ξ +

(S 2 − F1,2

)x1,2

The terms of the objective function may be explained referring to Table 4.1. The term n2 is the plannedpayments of the company, the term(p − S 2 − λ

)ξ is the profit of the company due to the project and(

S 2 − F1,2

)x1,2 is the transaction due to the futures contract. Finally, r

[y2

]− is the transaction due tofinancial distress cost incurred at time 3. Therefore, it is multiplied by β in order to derive the value ofit to time 2.

It can easily be observed that the firm will have a positive outcome from the futures contract if the priceof the derivative realizes higher than the price of futures contract, i.e. S 2 > F1,2. On the other hand,the firm will have a negative outcome if the opposite scenario comes true and this negativity may causefinancial distress. Indeed, in order to examine these situations, it is already known that the problem canbe shown as the following by multiplying the whole expression by the risk neutral probability densityfunction of random variable S 2:

V2 (y1) = minx1,2

∞∫0

[−4V2 (y1)

]φQ

S 2(S 2) dS 2

As observed from the equation above that the only term in the equation above that is effected by deci-sion x1,2 is the financial distress cost premium. The decision variable has no effect on the magnitudeof the term y1 − n2 +

(p − S 2 − λ

)ξ and the expected value of the term

(S 2 − F1,2

)x1,2 is equal to

zero. Thus, the first derivative of the objective function will be equal to the derivative of the financialdistress cost at t = 2 with respect to x1,2. Indeed, by minimizing the financial distress cost at t = 2, thedecisionmaker maximizes the expected value of the company.

Theorem 1: The optimum hedging decision of the company at time 1 for one-period model is x∗1,2 = ξ.

In this model, the only risk of facing financial distress is because of the price risk of the commodity.The price risk is completely eliminated by applying the full-hedging decision, i.e. the company shouldhedge at the same quantity with amount used for the assets delivered at the end of the project. Theresult approves the deductions in the study of Tanrisever, Duran and Sumer (2011). The details of theproof can be found in Appendix.

The solution found in this part may also be applied to forward contracts as well as futures (whenmargin call updates are neglected). On the other hand, full-hedging decision is especially suggested

14

to companies running their business with little amount of cash because the expected financial distresscost would be relatively high in such cases.

4.2 Two-Period Model

In this scenario where the firm signs a contract at time 1 for a delivery at time 3; and the companyreceives the revenue pξ from the contract and pays to the subcontractor at an amount of

(S 3 + λ

)ξ at

time 3. Similar to the previous model, the firm has a debt of ni that is planned to be paid at time t = iwhere i = 2, 3. Besides, the company has an initial cash reserve y1 and the cash balance of the firmat time i after the transactions are realized is again denoted as yi where i = 2, 3. It can also face afinancial distress costs of r

[yi]− at time 2 and 3, if the net cash balance of the firm decreases to a value

lower than zero. It is assumed that simple interest is used for calculation of the loans due to financialdistress. Additionally, the firm is assumed to have an overdraft contract with a bank. In this model,financial distress costs during the project are considered and they are collected (or at least accounted)together at time 4.

The firm considers to long x1,3 units of futures contracts at time 1 maturing at time 3 for the futuresprice F1,3, and x2,3 units of futures contracts at time 2 maturing at time 3 for the futures price F2,3. Bythese actions, the firm aims to decrease the risk of having financial distress cost at time 2 and time 3 dueto unstable price of the underlying commodity. In this part of the study, it is assumed that the marginaccount of the firm is updated at time 2. The amount of cash transactions at time 2 led by hedging aredenoted by

(F2,3 − F1,3

)x1,3. Besides, it is assumed that the price of the futures contract maturing at a

certain time is equal to the expected spot price of the underlying derivative at that time. Therefore atthe end of the futures contract, the account of the firm is updated by addition of

(S 3 − F2,3

)x2,3. Figure

4.2 shows all cash transactions of the company in two-period problem.

Figure 4.2: Summary of financial transactions for two-period model

To sum up, the firm has only two available decisions, it is the setting of hedging quantity at the at time1 and 2. The company may change the amount of the futures contract on hand without any additionaltransactions because the prices are evaluated at the end of every period and initial margin payment isneglected in the model. In addition, it has to borrow money with interest rate r if the cash balancebecomes negative. All financial transactions, decisions and actions of the company are shown togetherat Table 4.2.

15

Table 4.2: Summary of the events for two-period model

As observed from the table above, the decisionmaker should optimize hedging decisions at time 1and 2 by using the function V2 (y1). Since V2 (y1) includes the changes in the expected value of thecompany in the future, V3 (y2) should also be included in this function after multiplied by β. Afterconsidering all these facts and assumptions, the optimum hedging decision problem of the firm can besummarized as the following:

Objective Function at t = 1:

V2 (y1) = maxx1,34V2 (y1) = min

x1,3(−4V2 (y1))

= minx1,3

EφF2,3

[n2 +

(F1,3 − F2,3

)x1,3 − βV3(y2)

]

subject to:y2 = y1 − n2 +

(F2,3 − F1,3

)x1,3

where:

V3 (y2) = maxx2,34V3 (y2) = −min

x2,3(−4V3 (y2))

= minx2,3

Eφs3

[n3 −

(p − S 3 − λ

)ξ +

(F2,3 − S 3

)x2,3 + βr

([y2

]−+

[y3

]−)]

subject to:y3 = y2 − n3 +

(p − S 3 − λ

)ξ +

(S 3 − F2,3

)x2,3

It is already mentioned that the decisionmaker aims to maximize the expected total value of the com-pany at the end of the contract, t = 3. Therefore a dynamic programming approach is an appropriateway to deal with the problem. Firstly, the problem at time 2 should be solved. According to the previ-ous section which examines the decision for one-period model, the optimum hedging decision of the

16

decisionmaker at time 2 is x∗2,3 = ξ. In order to solve the optimization problem at t = 1, the solution ofthe first period should be placed into function V2 (y1), i.e. the decision variable x2,3 in function V2 (y1)should be replaced by ξ. Moreover, it is already known that the objective function should be multipliedby the risk neutral probability density functions of the variables F2,3 and S 3 to examine the expectedvalue of it.

Lemma 1: The optimum hedging decision of the company at t = 2 for two-period model is full hedging,i.e. x∗2,3 = ξ.

By using Theorem 1, this result is obvious for the company. Unless the magnitude of the ξ changes,the optimum hedging quantity at time 2 stays at the same level.

Theorem 2: The optimum hedging decision of the company at t = 1 for the two-period model is0 5 x∗1,3 5 ξ.

The details of the solution can be found in the Appendix. This result proves that decisionmakers mayunder-hedge (in some special cases full hedge or no hedge) independent of any parameters if there isan opportunity to update the hedging amount before the project ends. This outcome is logical becausecompany needs to balance two factors that can create financial distress while it can still update itsdecision in further periods.

• Risk Factor I: The first risk is facing financial distress in the project delivery time due to changingprice of the commodity. In order to minimize this risk, the firm needs to hedge at an amount ofξ. If the amount becomes less or more than ξ, the risk of financial distress appears.

• Risk Factor II : The second factor is emergence of financial distress cost before the projectdelivery time, which is caused by margin call updates. In case of a decrease in the price of thefuture contract and the company has long position, the company needs to deposit margin callupdates; and the same situation occurs in short position when price increased. However, if thehedging amount is too high, the company may face a shortage of cash to pay deposits. Therefore,in order to minimize this risk, the firm should not hedge at any amount.

In conclusion, one risk factor is minimized at full hedging decision and the other is minimized at nohedging. Thus, it is obvious that the optimum hedging amount will be between 0 and ξ but the exactoptimum solution depends on the parameters in the model. The details of the proof can be found inAppendix.

For ease of notation, two terms are defined. 4 denotes the future cash reserve of the company at time2 when the company does not hedge at all and it is equal to y1 − n2. Besides, δ is used to describe theexpected profit margin of the company per unit and it is calculated as p − λ − F1,3.

Theorem 3: The optimum hedging decision at t = 1 is x∗1,3 = ξ4/ (24 − n3 + δξ) when4 (4 − n3 + δξ) =0, and x∗1,3 = −ξ4/ (−n3 + δξ) when 4 (4 − n3 + δξ) 5 0 satisfying that 0 5 x∗1,3 5 ξ.

The result is found by using the features of normal distribution, and true for any kind of centralweighted symmetrical distributions. As well as theoretical findings, it may also be used as a bene-ficial approach in real life problems in case of roughly symmetrical price distribution assumptions. Italso provides an opportunity for sensitivity analysis with respect to different values of the variables inthe model. The details of the proof can be found in Appendix. The final solution can be summarizedas the following.

17

x∗1,3 =

ξ4/ (24 − n3 + δξ) i f 4 (4 − n3 + δξ) = 0

−ξ4/ (−n3 + δξ) i f 4 (4 − n3 + δξ) 5 0

Theorem 4: The optimum hedging decision of the company at t = 1 when 4 = 0 and 4 , n3 − δξ fortwo-period model is not hedging any unit of the underlying commodity, i.e. x∗1,3 = 0.

If the company will have a zero cash balance at t = 2 without hedging, then the cash balance of thecompany may decrease below zero and hedging could cause a financial distress cost (at %50 per centprobability for symmetrical distributions). In order to prevent a financial distress cost at the end of firstperiod, the company should not hedge at all. The details of the proof can be found in Appendix.

Theorem 5: In case of 4 = n3 − δξ and 4 , 0, the optimum hedging decision at t = 1 is x∗1,3 = ξ .

Expected cash balance of the company at the end of the project without hedging payment is 0 in theseconditions. As a result, the cash balance of the company may decrease below zero and hedging couldcause a financial distress cost (at %50 per cent probability for symmetrical distributions). In order toprevent a financial distress cost at the end of the project, the company should hedge at amount of ξ. Inconclusion the optimum decision is x∗1,3 = ξ. The details of the proof can be found in Appendix.

Theorem 6: In case of 4 = n3 − δξ = 0 and 4 = 0, the hedging decision becomes insignificant while0 5 x∗1,3 5 ξ, in other words it does not effect to the expected financial distress cost and all decisionssatisfying 0 5 x∗1,3 5 ξ are optimum.

In this case, the company will have a zero cash balance at t = 2 without hedging, then the cash balanceof the company may decrease below zero and hedging could cause a financial distress cost. In orderto prevent a financial distress cost at the end of first period, the company should not hedge at all.On the other hand, expected cash balance of the company at the end of the project without hedgingtransactions is 0 in these conditions. Therefore, the optimum decision to minimize this risk is fullhedging.

In conclusion, both of the factors that affects to the hedging decision of the company are at their highestinfluence points. Besides, both have the same effect in magnitude but one tries to push the decisionto the level 0 while the other pulls it to to ξ. Therefore, all hedging decisions in the interval

[0, ξ

]are

mathematically optimum in these conditions. The details of the proof can be found in Appendix. I

Theorem 7: In case of 4 approaching plus or minus infinity, i.e. 4 → ±∞, the optimum hedgingquantity line approaches to ξ/2. However, hedging lose its significance after some point and becomesa value-neutral action due to the probability density function of the normal variable F1,2.

In case that the firm has a lot of cash, it does not need to hedge its money because there is no riskof having financial distress cost in this condition. Similarly, if its cash reserve is hugely negative, itcannot prevent the financial distress cost by hedging; as a result it would not be necessary to hedge atany amount. The details of the proof can be found in Appendix.

Theorem 8: In case of n3 approaching positive or minus infinity, i.e. n3 → ±∞, the optimum hedgingquantity line approaches to 0. However, multi-optimality occurs due to the probability density functionof the normal variable F1,2.

In case that the firm will have a lot of cash at t = 3, it does not need to hedge its money because thereis no risk of having financial distress cost in this condition. Similarly, if its cash reserve is hugely

18

negative, it cannot prevent the financial distress cost by hedging, as a result it would not be necessaryto hedge at any amount. The details of the proof can be found in Appendix.

Summary of theorems at two-period model:

After all the examinations completed in this section, hedging quantity is mainly dependent on the valueof 4 and expression n3 + δξ. By analyzing the values of these variables, an appropriate decision canbe made by the decisionmaker. As well as the outcomes in the Table 4.3, It is also true that hedgingbecomes a value neutral action while 4 or n3 is getting closer to ±∞. The summary of theorems in thissection are shown in Table 4.3.

Table 4.3: Summary of theorems for two-period model

4.3 Three-Period Model

The company signs a project contract at t = 1. It receives a revenue of pξ from the customers when theproject is completed at t = 4. On the other hand, it pays the cost of assets at an amount of

(S 4 + λ

)ξ

to the subcontractor at the end of the project, again at t = 4. Additionally, the firm has debts of ni

that is planned to be paid at t = i where i = 2, 3, 4. The cash balance of the firm at time i afterthe transactions are realized is denoted as yi where i = 1, 2, 3, 4. Obviously, the company may facefinancial distress costs when the net cash balance of the firm decreases to a value lower than zero attime i, where i = 2, 3, 4 . In this model, accumulated financial distress costs during the project are paidor accounted at t = 5 and they are desired to be minimized.

The firm considers to long xi,4 units of futures contracts at time i maturing at t = 4 for the futures priceFi,4 where i = 1, 2, 3. By hedging, the firm aims to decrease the risk of having financial distress costs,r[yi]−, at i = 2, 3, 4 due to the unstable price of the underlying commodity in its project. In this part of

the study, it is assumed that the margin account of the firm is updated at t = 2 and t = 3. The amountof cash transactions at time i led by hedging are denoted by

(Fi,4 − Fi−1,4

)xi−1,4 where i = 2, 3. Due to

the previously mentioned price assumptions of the futures contracts and spot prices of the commodity,it is assumed that F1,4 = Eφ

(F2,4

)= Eφ

(F3,4

)= Eφ

(S 4

)for a decisionmaker at t = 1. The decision

variable at t = i is the amount of the units of futures contract undertaken at t = i for maturity datet = 4 which is denoted as xi,4 where i = 1, 2, 3. All financial transactions of the described problem arebriefly shown at Figure 4.3.

19

Figure 4.3: Summary of financial transactions for three-period model

Briefly, there are four time periods, as a result five different time epochs for events. The firm hasto borrow money from the bank if cash balance goes to negative. Therefore, the company tries tominimize its expected financial distress cost by using three different decision variables. A clear andbrief view of the information on the problem is shown in Table 4.4.

Table 4.4: Summary of the events for three-period model

As observed from the table above, the decisionmaker should optimize hedging decisions at three dif-ferent time epochs, time 1, 2 and 3. Similar to the previous models, in order to achieve this aim, thecompany use the function V2 (y1), the maximum value of the expected the future cash change of thecompany at t = 1 which is discounted to the respective value of t = 2. After considering all these factsand assumptions, the optimum hedging decision problem of the company at t = 1 can be summarizedas:

V2 (y1) = maxx1,34V2 (y1) = min

x1,3(−4V2 (y1))

= minx1,4

EφF2,4

[n2 +

(F1,4 − F2,4

)x1,4 − βV3(y2)

]20

subject to:y2 = y1 − n2 +

(F2,4 − F1,4

)x1,4

where:

V3 (y2) = maxx2,4

EφF2,4

[−n2 +

(F3,4 − F2,4

)x1,4 + βV4(y3)

]

subject to:y3 = y2 − n3 +

(F3,4 − F2,4

)x1,4

and:

V4(y3) = maxx3,4

Eφs4

[−n4 +

(p − S 4 − λ

)ξ +

(S 4 − F3,4

)x3,4 − βr

([y2

]−+

[y3

]−+

[y4

]−)]

subject to:y4 = y3 − n4 +

(p − S 4 − λ

)ξ +

(S 4 − F3,4

)x3,4

Lemma 2: The optimum hedging decision of the company at t = 3 for three-period model is full-hedging, i.e. x∗3,4 = ξ.

This result follows from Theorem 1. Unless the magnitude of the ξ does not change, the optimumhedging quantity at t = 3 stays at the same level.

Lemma 3: Suppose that the terms 4′

and δ are defined such as 4′

= y2 − n3 and δ = p − λ − F2,4.The optimum hedging decision at t = 1 is x∗2,4 = ξ4

′

/(24

′

− n4 + δξ)

when 4′(4′

− n4 + δξ)= 0, and

x∗2,4 = −ξ4′

/ (−n3 + δξ) when 4′(4′

− n4 + δξ)5 0 satisfying that x∗2,4 5 ξ.

The result directly follows from Theorem 2. On the other hand, both y2 and δ has stochastic term F2,4

inside and it means that the decision at x∗2,4 is dependent on the realized value of the futures contract.

Theorem 9: The optimum hedging decision of the company at t = 1 for three-period model is 0 5x∗1,4 5 x∗2,4.

After examinations of all possible cases with respect to the decision variables, the optimum policy isfound as underhedging and details of the solution may be found in Appendix.

The interpretation of Theorem 9 is that the decisionmaker should increase its hedging amount whilegetting closer to the delivery date in an optimum policy, such that 0 5 x∗1,4 5 x∗2,4 5 x∗3,4 = ξ. However,it should be noted that this problem is a dynamic problem and after the futures prices in a period isrealized, it has to be solved again for the next period.

Moreover, Theorem 9 can be generalized for the cases where the decisionmaker has opportunities toupdate its hedge amount more than three times, in Theorem 10.

Theorem 10: The optimum hedging decision of the decisionmaker at t = 1 for n period model isalways underhedging and is less than or equal to the expected hedging amount of the next period, i.e.0 5 x∗1,n 5 x∗2,n 5 ... 5 x∗n−2,n 5 x∗n−1,n = ξ.

After examinations of all possible cases with respect to the decision variables, the optimum policy isfound as underhedging and details of the proof may be found in Appendix.

21

The interpretation of this theorem is the following. The decisionmaker should increase its hedgingamount while getting closer to the delivery date in an optimum policy, such that 0 5 x∗1,n 5 x∗2,n 5... 5 x∗n−2,n 5 x∗n−1,n = ξ; since the decisionmaker has more flexibility to update the hedge amountfurther from the delivery date. As mentioned earlier, it is an important point that this problem is adynamic problem and after the random variables in a period are realized, it has to be solved again forthe next period. In case of a false decision that is made during a period, the decisionmaker may justifythe optimum hedging quantities in the beginning of the periods again and should be able to increase ordecrease the hedging quantity. In such a case, the inequality stated in Theorem 10 may not be satisfied;however the reason of it is the false hedging decisions during the previous periods.

Theorem 11: Similar to two-period problem, hedging is becoming a value neutral action when 4 goesto ±∞ in multiperiod model.

When the company has a lot of cash, it is not necessary to hedge at time 1. As long as facing financialdistress is out of possibility, hedging at reasonable amounts does not affect its expected cash balancein the future. Consequently, hedging becomes a value neutral action as long as the company hedge atreasonable amounts. This amount is affected by the magnitude of 4 and standard deviation σ.

4.4 Approximation of Two-Period Model with Compound Interest

As mentioned earlier, this paper mainly aims to examine how to avoid financial distress cost undersimple interest. However, compound interest is also charged by financial institutions in real life andthe results found in two-period simple interest model may be used as an approximation to solution ofthe same problem with compound interest. By the simple interest approximation, we can provide asolid decision methodology for real life decisionmakers using compound interest rates.

The mathematical model is almost the same with simple interest except that the financial distress costsare directly accumulated at the period it is incurred. Therefore, interest of financial distress cost isalso accounted in the model. After considering all these facts and assumptions, the optimum hedgingdecision problem with compound interest rates for a two-period problem may be summarized as thefollowing:

V2 (y1) = maxx1,34V2 (y1) = min

x1,3(−4V2 (y1))

= minx1,3

EφF2,3

[n2 +

(F1,3 − F2,3

)x1,3 + r

[y2

]−− βV3(y2)

]

subject to:

y2 = y1 − n2 +(F2,3 − F1,3

)x1,3

where:

V3 (y2) = maxx2,34V3 (y2) = −min

x2,3(−4V3 (y2))

= minx2,3

Eφs3

[n3 −

(p − S 3 − λ

)ξ +

(F2,3 − S 3

)x2,3 + βr

[y3

]−]22

subject to:y3 = y2 − n3 +

(p − S 3 − λ

)ξ +

(S 3 − F2,3

)x2,3 − r

[y2

]− .The company tries to minimize the total of expected financial distress costs discounted to time 2 whichis formulated as r

([y2

]−+ β

[y3

]−). However, the most critical observation to make in this formulationis the fact that the term

[y3

]− includes[y2

]− inside. This makes the problem harder to solve by classicalmathematical approaches.

Lemma 4: The optimum hedging decision of the company at t = 2 for two-period model is full hedging,i.e. x∗2,3 = ξ.

In one-period problem, there is no difference in the optimum hedging quantity because there is nodifference between compound and simple interest rates formulation for one-period. In addition, thecompany is again trying to minimize the expected value of the only possible financial distress cost inthe future. Therefore, interest of interest is not considered and the optimum hedging decision at time2 is equal to the quantity of commodity used for the project. When this result is used in the model,further calculations are possible.

Theorem 12: The optimum hedging decision of the company at t = 1 for two-period model is 0 5x∗1,3 5 ξ.

Similar to the model with simple interest, the optimum hedging decision at time 1 is also between 0and ξ. The details of the proof can be found in Appendix.

Theorem 13: The optimum hedging decision of the company at t = 1 when 4 = y1 − n2 = 0 fortwo-period model with compound interest is not hedging any units at all, i.e. x∗1,3 = 0.

Similar to the model with simple interest, the cash balance of the company may decrease below zeroand hedging could cause a financial distress cost (at %50 per cent probability for symmetrical distri-butions) if the company will have a zero cash balance at t = 2 without hedging. In order to prevent afinancial distress cost at the end of first period, the company should not hedge at all. The details of theproof can be found in Appendix.

Theorem 14: The optimum hedging decision of the company at t = 1 cannot be higher than ξ/ (1 + r)when 4 < 0. The optimum hedging quantity will be in the interval of 0 5 x∗1,3 5 ξ/ (1 + r).

If the net cash reserve of the company will be less than zero at time 2 except the transactions due tohedging, then the maximum optimum hedging amount of the decisionmaker is ξ/ (1 + r). Otherwisethe firm can face a higher expected financial distress cost due to hedging and the expected net cashreserve of the company will decrease at t = 1. The details of the proof can be found in Appendix.

Although the model with compound interest behaves similar to the model with simple interest for two-period, finding mathematical results for three or more periods is not as easy as the model with simpleinterest. Therefore, our idea is to use simple interest model as an approximation to this model and usenumerical data analysis to judge the goodness of this approximation.

4.5 One-Period Model with Call Option

This case is assumed to be a scenario where there are no financial transactions between sign of thecontract and delivery of project. The firm has a debt at the amount of n2 that is planned to be paid at

23

time 2. It will receive the revenue from customer at an amount of pξ and will have to pay the cost ofthe assets to the subcontractor at an amount of (S 2 + λ) ξ also at time 2. The cash balance of the firmat the beginning of the project is y1.

In the beginning of the period, the firm considers to long x1,2 units of call options on the commodityfutures at time 1 maturing at time 2 for the strike price F1,2, which is the expected spot price of thecommodity at time 2. By this action, the firm aims to decrease the risk of having financial distress costat time 2 due to unstable price of the commodity. Unlike the using of future contract to hedge, thistime the firm needs to pay risk premium to buy the contract at time 1 and the unit price of the optioncontract is equal to h where, h =

∫ ∞F1,2

(S 2 − F1,2

)φQ

S 2(S 2) dS 2. By this action, the firm aims to decrease

the risk of having financial distress cost at time 2 due to unstable price of commodity. It is assumedthat the account of the firm is only updated at time 2. The amount of cash transaction at time 2 led byhedging is denoted by

(S 2 − F1,2

)+x1,2. It means that the firm will get paid by

(S 2 − F1,2

)x1,2 when

realized spot price of the commodity at time 2 is higher than the strike price of the option contract,which is F1,2 for our model. Otherwise, there will be no financial transactions at time 2 due to theoption contract. The only decision variable for the firm is the amount of call options to buy at time 1for maturity date time 2, which is denoted as x1,2.

Unlike the other models considered in this study, the firm is assumed to pay a risk premium to buythe financial derivative. Therefore, y

′

1 is defined the cash balance of the firm after buying the optioncontracts and it is formulated as y

′

1 = y1 − hx1,2.

All financial transactions of the described problem are briefly shown at the Figure 4.4.

Figure 4.4: Summary of financial transactions for one-period model with call option

To sum up, the company has only one available decision, the hedging quantity at the at time 1. Besidesit has to borrow money with interest rate r if the cash balance becomes negative. All financial transac-tions, decisions and actions of the company are shown together at the summary of events table below,at Table 4.5.

24

Table 4.5: Summary of events for one-period model with call option

Again the company aims to maximize the expected total value of the company at the end of the project.Therefore, V j (yi) is used as the objective function which maximizes the expected change in the valueof the company at t = j with an initial cash yi. Besides another function 4Vi+1 (yi) is defined todescribe the change in the expected value of the company in period i with an initial cash yi. Thus, theobjective function V j (yi) can be defined as a function referring the maximum value of the function4Vi+1 (yi), which can be translated into a minimization problem by minimizing −4Vi+1 (yi). Sincethe only decision variable is x1,2, the decision problem faced by the company can be summarized asfollows:

V2 (y1) = maxx1,24V2 (y1)

= minx1,2− 4V2 (y1)

= minx1,2

Eφs2

[n2 −

(p − S 2 − λ

)ξ + hx1,2 −

[S 2 − F1,2

]+x1,2 − r

[y′

1

]−+ βr

[y2

]−]Subject to:

y2 = y1 − n2 +(p − S 2 − λ

)ξ − hx1,2 +

[S 2 − F1,2

]+x1,2

y′

1 = y1 − hx1,2

where

h =

∞∫F1,2

(S 2 − F1,2

)φQ

S 2(S 2) dS 2

25

The terms of the objective function may be explained referring to Figure 4.5. The term hx1,2 is the costof buying or selling call options, n2 is the planned payments of the company, the term

(p − S 2 − λ

)ξ is

the profit of the company due to the project and(S 2 − F1,2

)+x1,2 is the possible gain from call options.

Finally, r[y2

]− is financial distress cost incurred at time 3. Therefore, r[y2

]− is multiplied by β in orderto discount it to time 2.

The only term in the objective function affected by the decision of x1,2 is the financial distress costpremium. The decision variable x1,2 has no effect on the magnitude of the term y1−n2 +

(p − S 2 − λ

)ξ

and the expected value of the term(S 2 − F1,2

)x1,2 is equal to zero. Thus, the first derivative of the

objective function will be equal to the derivative of the financial distress cost at t = 2 with respectto x1,2. Indeed, by minimizing the financial distress cost at time 2, the decisionmaker maximizes theexpected value of the company.

Theorem 15: The optimum amount of call options to buy at time 1 with the strike price F1,2 andmaturity date 2 is always less than or equal to y1

h .

The company buys some call options to prevent financial distress costs at t = 2 by paying the expectedpayoff. However, if the firm pays expected payoff more than its budget at t = 1, it will immediatelyface financial distress cost instead of avoiding it. As a result, it should not buy more call options thanits budget at t = 1. The details of the proof can be found in the Appendix.

When the result of Theorem 15 is utilized, the overall model can be rewritten:

V2 (y1) = maxx1,24V2 (y1)

= minx1,2− 4V2 (y1)

= minx1,2

Eφs2

[n2 −

(p − S 2 − λ

)ξ + hx1,2 −

[S 2 − F1,2

]+x1,2 + βr

[y2

]−]

Subject to:

y2 = y1 − n2 +(p − S 2 − λ

)ξ − hx1,2 +

[S 2 − F1,2

]+x1,2

x1,2 <y1

h

where

h =

∞∫F1,2

(S 2 − F1,2

)φQ

S 2(S 2) dS 2

Theorem 16: y1h +

((p−λ−F1,2)ξ−n2

h

)+

= x∗1,2 = ξ as long as y1h +

((p−λ−F1,2)ξ−n2

h

)−is more than ξ.

The details of the proof may be found in Appendix. Supporting numerical results may also be foundin Chapter 5, Numerical Analysis.

26

When y1h +

((p−λ−F1,2)ξ−n2

h

)−< ξ, an optimal as suggested by Theorem 16 is not possible. In such

situations, the company may intuitively try to equalize its hedging level to y1h +

((p−λ−F1,2)ξ−n2

h

)−in order

to get closer to the optimality due to convexity. Although we do not prove this claim mathematicallyin this study, it will also be examined in numerical analysis part of this study.

27

CHAPTER 5

NUMERICAL ANALYSES

Numerical analysis is performed to observe the effects of mathematical results and gain further insightthat could not be acquired by formal mathematical analysis. Simulation models are utilized in numer-ical analysis, and random variables are created by suitable random number generators for the relateddistributions. For all decision settings, the same random price streams are used and financial distresscosts are calculated accordingly.

In mathematical models used throughout this study, futures contract prices are assumed to be stochas-tic. Indeed F j+1,i is assumed to follow a normal distribution with mean F j,i and standard deviationσ and the spot price of the underlying commodity S i has a mean of Fi−1,i and standard deviation σ.Therefore, the realized value of the futures contract at a period is used as the mean of the futurescontract price at the next period maturing at the same date in numerical analysis.

In order to eliminate the bias of random number generator, a specific approach is used. It is alreadystated that F1,i is the mean and σ is the standard deviation for the price of futures contract F2,i atthe beginning of the project. These variables are utilized in the normal distribution random numbergenerator and necessary random numbers for futures prices in the model are created. Suppose thatm random variables are generated for period 2 (in other words, random F2,i values are created form times) by using the generator, and jth number is referred as F( j,+)

2,i . After we create F( j,+)2,i using

the random number generator with the mean of F1,i, another number F(1, j,−)2,i is calculated such that

F( j,+)2,i + F( j,−)

2,i = 2F1,i. Thus, totally 2m numbers are created denoted as F( j,Ω1)2,i and where Ω1 may be

either (+) or (−) and 1 5 j 5 m .

For the next period, this time the numbers F( j,Ω1)2,i are placed into the random number generators as

mean of the price at the next period, and 2m numbers are generated denoted as F( j,Ω1,+)3,i . By using these

numbers, the numbers F( j,Ω1,−)3,i are also created such that F( j,Ω1,+)

3,i + F( j,Ω1,−)3,i = 2F( j,Ω1)

2,i where Ω1 may

either be + or − and 1 5 j 5 m. As a result, 4m numbers are obtained as F( j,Ω1,Ω2)3,i for the future price

of the second period where Ω1 and Ω2 may either be (+) or (−) and 1 5 j 5 m. The random numbergeneration procedure is illustrated below for the first two-periods.

F1,i

F2,i = F( j,+)

2,i

F3,i = F( j,+,+)3,i

F3,i = F( j,+,−)3,i

F2,i = F( j,−)2,i

F3,i = F( j,−,+)3,i

F3,i = F( j,−,−)3,i

By this procedure, 4m different price sets are composed for the first two-period, and the overall means

29

of both of the random stages are equal to F1,i. The price sets of futures contracts for t = 1, 2, 3 are asthe following where j is an integer such that 1 5 j 5 m.

(F1,i, F

( j,+)2,i , F( j,+,+)

3,i

)(F1,i, F

( j,+)2,i , F( j,+,−)

3,i

)(F1,i, F

( j,−)2,i , F( j,−,+)

3,i

)(F1,i, F

( j,−)2,i , F( j,−,−)

3,i

)As stated before, all sets should satisfy F( j,+)

2,i + F( j,−)2,i = 2F1,i and F( j,Ω1,+)

3,i + F( j,Ω1,−)3,i = 2F( j,Ω1)

2,i forΩ1 = +,−. All numerical analyzes in this study are completed by using this procedure with 100,000sample data size.

During numerical analysis, multi-optimality is observed as Theorem 7 and 8 suggested. The multi-optimal decisions exits due to the normally distributed future prices and significance limit of the de-cisionmaker for the expected financial distress cost. According to the observations, multi-optimalityexists while the terms b and c are becoming far from the mean F1,3 with respect to the standard devia-tion σ. In order to analyze this phenomenon, let us define a magnitude ζ that can be neglected by thedecisionmaker in the objective function. In addition, a function D

(x1,3, α

)is defined as the following:

D(x1,3, α

)=

∣∣∣∣4V2

(y1|x1,3 = x∗1,3

)− 4V2

(y1|x1,3 = x∗1,3 + α

)∣∣∣∣Suppose that an optimum interval of x∗1,3 is identified such that[x∗1,3 + α−, x∗1,3 + α+

]where α+ and α− are integers as α+ > 0 > α−. It means that α+ is defined

as the positive deviation from the optimum decision x∗1,3 and α− is the negative deviation. Then thedeviations α+ and α− are calculated by solving D

(x1,3, α

+) D

(x1,3, α

−)= ζ. However, a linear

differential approximation may be applied by assuming that new line D(x1,3, α

+)∣∣∣∣∣∣α+(1/2)d

(4V2

(y1 |x1,3=x∗1,3+α+

)+4V2

(y1 |x1,3=x∗1,3

))dx1,3

∣∣∣∣∣∣. Sinced(4V2

(y1 |x1,3=x∗1,3

))dx1,3

is equal to 0, the function D(x1,3, α

+)

becomes the following:

D(x1,3, α

+) ∣∣∣∣∣∣α+ (1/2) d(4V2

(y1|x1,3 = x∗1,3 + α+

))dx1,3

∣∣∣∣∣∣ ζThen the value of α+ is approximately calculated as the following:

α+ 2ζ∣∣∣∣∣∣ d

(4V2

(y1 |x1,3=x∗1,3+α+

))dx1,3

∣∣∣∣∣∣5.1 Analysis of Two-Period Model

The variables 4 (cash balance of the company at t = 2 without hedging) and n3 (payment at t = 3)are the most important variables. Therefore, the behavior model will be examined as these two mainparameters are changing.

30

Figure 5.1: Expected total financial distress costs for x1,3 values for two-period model with base pa-rameter setting

A set of values of the parameters will be taken into account as base settings for numerical analysis. Inthis base setting, the values of the variables are chosen as y1 = 100, n2 = 30, n3 = 30, ξ = 100, p = 11,λ = 0.5, F1,3 = 10, σ = 1, r = 0.1, β = 0.95. Besides, amounts less then ζ = 0.0001 units of expectedfinancial distress cost is supposed to be insignificant for the company, as a result multi-optimality existfor the company when difference of financial distress costs with respect to different decisions are lessthan ζ = 0.0001. Expected total financial distress costs for the base case with respect to changingvalues of x1,3 are shown below. For each x1,3 decision in Figure 5.1, FDC represents the average totalfinancial distress cost for 100, 000 different realizations of the future and commodity price realization.As observed from Figure 5.1, the function is convex and the optimum decision for the model is 45which is less than ξ = 100 as Theorem 2 suggests.

5.1.1 Analysis of Two-Period Model With Changing Values of 4

4 = y1 − n2 is an effective parameter to the hedging decision and it is also important for real lifedecisionmakers. Moreover, while two-period model is examined mathematically in previous sectionsof this study, a summary of theorems table is already provided at Table 4.3 and it is primarily based onthe values of 4. Therefore, the first parameter to consider should be 4. Typical behavior of the optimaldecision for changing 4 values is obtained in Figure 5.2 when the model is solved with base settingparameters.

In Figure 5.2, X axis represents different 4 values while Y axis shows the optimum hedging deci-

31

Figure 5.2: Simulated optimum hedging decisions corresponding to changing values of 4 for two-period model with base parameters

sions for corresponding 4 values. In both extreme sides of X axis, multiple optimality is observed assuggested in Theorem 7 and the points where multi-optimality started to be observed will also be ex-amined in this section. On the other hand, firstly, the behavior of the optimal hedging quantity functionshould be inspected by using Theorem 3. Theorem 3 suggests the following result for solution of themodel:

x∗1,3 =

ξ4/ (24 − n3 + δξ) i f 4 (4 − n3 + δξ) = 0

−ξ4/ (−n3 + δξ) i f 4 (4 − n3 + δξ) 5 0

In order to check the conformity to the theoretical results obtained in previous settings, the expressionsξ4/ (24 − n3 + δξ) and −ξ4/ (−n3 + δξ) should also be drawn while 4 is changing and they will becalled as Line 1 and Line 2, respectively in Figure 5.3. Figure 5.3 shows the optimum hedging valuessuggested by Theorem 3 with corresponding 4 values. It is observed that the solution found by usingTheorem 3 is an exact fit to the solution found by simulation in Figure 5.3 except multi-optimumsolutions. Consequently, the multi-optimality conditions should also be examined.

As stated in Theorem 4, the optimum hedging amount x∗1,3 is equal to 0 when 4 = 0. On the otherhand, the optimum hedging amount is equal to ξ when 4 = n3 − δξ. Between these two points, theoptimum line is flat with a constant slope of ξ/ (n3 − δξ) which is shown as Line 2 at Figure 5.3.Outside this region, the optimum line has a curvy shape since the optimum hedging amount is equal toξ4/ (24 − n3 + δξ) and it is represented as Line 1 in Figure 5.3. Besides, the line is converging to ξ/2at both sides of X axis.

32

Figure 5.3: Theoretical optimum hedging decisions corresponding for different values of 4 for two-period model with base parameters

In order to analyze where multi-optimality starts while 4 → ±∞, b and c should carefully be analyzed.It is already known that Line 1 is the optimum line and b = c = F1,3−(24 − n3 + δξ) /ξ while 4 → ±∞.As a result, the point where multi-optimality begins (such as the decision x∗1,3 and x∗1,3 + 1 will beindifferent for the decisionmaker) is the point that satisfy D

(x1,3, 1

)= ζ. Although, the first derivative

of the expected total financial distress cost with respect to x1,3 is hard to calculate, we may use thelinear differential approximation mentioned in the beginning of this section in order to understand thevariables affecting to the multioptimality.

Suppose that, 4+ is the value of 4 where multi-optimality starts while 4 → +∞. It is known that theoptimum Line 1 (which represents x∗1,3 = ξ4/ (24 − n3 + δξ)) is symmetrical with respect to the point(4 = (−n3 + δξ) /2, x∗1,3 = ξ/2

), and this situation is also observed at Figure 5.3. Therefore, the starting

point of multi-optimality while 4 → −∞may be defined as 4−such that 4++4− = n3−δξ. On the otherhand, it is numerically observed that the one of the most important factor on where multi-optimalitybegins is standard deviation σ of futures prices of the commodity due to characteristic behavior ofnormal distribution. However, standard deviation σ does not affect magnitude of the optimum decisionvalue, as observed in Theorem 3.

Now analysis of the total expected financial distress cost will be continued by considering other dis-crete parameters while 4 continuously changes. In these examinations, multi-optimality region will beignored in the figures in order to improve the presentation.

33

Figure 5.4: Optimum hedging decisions corresponding to changing values of 4 for two-period modelwith base parameters and different n3 levels

5.1.1.1 Effect of n3 while 4 values change:

In order to analyze the effect of n3, again especially Theorem 3 should be considered. Therefore, thepoint where 4 = n3 + δξ = 0 is important and this is satisfied by n3 = 50 when predefined base valuesof the parameters are used. As stated in Theorem 8, there is multi-optimality while we get closer tothe origin, 4 = 0 when n3 = 50.

Five different values of n3 are utilized such as 10, 30, 50, 70, 90 while the other parameters are setto their predefined base values. Figure 5.4 shows the respective optimum hedging decisions while4 changes on a continuous interval and n3 has these values. In addition, the optimum hedging linewhen n3 = 50 is shown by two different lines to show the multi-optimal area. Moreover, in case ofi + j = 2 ∗ 50, the graphical representations of the optimum hedging lines with n3 = i and n3 = j willbe symmetrical to each other with respect to Y axis. For example, the optimum value lines where theparameter value n3 is 30 and 70 are symmetrical with respect to Y axis. As obtained from Theorem 5and observed from Figure 5.4, the point where the optimum hedging quantity reaches to ξ shifts withdirect proportion to value of n3. Consequently, n3 is a vital parameter for setting the hedging amountand decisionmakers should use approximations to set their hedging position on the optimum line inFigure 5.4.

34

Figure 5.5: Optimum hedging decisions corresponding to changing values of 4 for two-period modelwith base parameters and different δ levels

5.1.1.2 Effect of δ while 4 values change

The value of the expected profit margin δ = p − λ − F1,3 is another important factor for the companywhile setting the hedging level. The value of δ should be positive, to get the company involved in theproject. Similar to the analysis of the effect of n3, the point where 4 = n3 + δξ = 0 is important andthis is satisfied by δ = 0.3 when predefined base values of the parameters are used. Furthermore, againmulti-optimality is observed as getting closer to origin, 4 = 0 when δ = 0.3.

Five different values of δ are exercised such as 0.1, 0.3, 0.5, 0.7, 0.9 while the other parameters are setto their predefined base values. Figure 5.5 shows the respective optimum hedging decisions while 4changes and δ has these five value levels. Besides, the optimum hedging line when δ = 0.3 is shown bytwo different lines to show the multi-optimal area. The graphical representation of optimum hedginglines when value of the parameter δ is equal to i and j will be symmetric to each other with respect to Yaxis as long as i + j = 2 ∗0.3. For example, the optimum value lines where the parameter value δ is 0.1and 0.5 are symmetric with respect to Y axis. As obtained from Theorem 5 and observed from Figure5.5, the point where the optimum hedging quantity reaches to ξ shifts with direct proportion to valueof δ. Consequently, δ is another vital parameter for setting the hedging amount and decisionmakersshould use approximations to set their hedging position on the optimum curve in Figure 5.5.

35

Figure 5.6: Optimum hedging decisions corresponding to changing values of 4 for two-period modelwith base parameters and different ξ levels

5.1.1.3 Effect of ξ while 4 values change

The value of the quantity of the assets required for the project ξ is a vital factor for the company to setits hedging position. Referring to Theorem 2, it is seen that the value of ξ determines the boundariesof the optimum region. Besides, Theorem 3 suggests that it also affects the slope of the lines ofoptimum solution while 4 changes. Similar to the analysis of the effect of n3 and δ, the point where4 = n3 + δξ = 0 is important and this is satisfied by ξ = 60 when predefined base values of theparameters are used. Therefore, multi-optimality emerges while getting closer to origin, 4 = 0 whenξ = 60 .

Four different value levels of ξ are exercised; 20, 60, 100, 140, while the other parameters are set totheir predefined base values. Figure 5.6 shows the respective optimum hedging decisions while 4changes and δ has these 4 value levels. Moreover, the optimum hedging line when ξ = 60 is shown bytwo different lines to show the multi-optimality region.

Since the magnitude of ξ has a direct proportion with the risk of financial distress due to price risk, itis an effective parameter. As mentioned above, the upper limits of hedge quantity is also determinedby the value of ξ as observed from the curve of optimum hedge decision in Figure 5.6.

The following facts can be summarized as important for of real life decisionmakers:

• The optimum hedging quantity always follows the line ξ4/ (n3 − δξ) starting from origin pointuntil reaching to ξ where 4 = n3 − δξ.

• When the expression n3 − δξ is negative and multi-optimal region is ignored, the optimum hedg-ing decision at t = 1 is always higher than ξ/2 as long as 4 < n3 − δξ. On the other hand, it is

36

Figure 5.7: Optimum hedging decisions corresponding to changing values of 4 for two-period modelwith base parameters and different n3 levels

always lower than ξ/2 as long as 4 > 0 for this case.The value of n3−δξ with base parameter values except n3 = 10 is equal to 40. Therefore, the linerepresenting the optimum hedging decisions when n3 = 10 in Figure 5.7 is an example for thecase described above. The optimum hedging line follows the line ξ4/ (n3 − δξ) in the interval[n3 − δξ, 0

]; additionally it is always higher than ξ/2 in the interval

[−∞, n3 − δξ

]and always

lower than ξ/2 in the interval [0,+∞].• When the expression n3 − δξ is positive and multi-optimal region is ignored, optimum hedging

decision at t = 1 is always higher than ξ/2 when 4 > n3 − δξ. On the other hand, it is alwayslower than ξ/2 as long as 4 < 0 for this case.The value of n3 − δξ with base parameter values except n3 = 90 is equal to −40. Therefore,the line representing the optimum hedging decisions when n3 = 90 in Figure 5.7 is an examplefor the case described above. The optimum hedging line follows the line ξ4/ (n3 − δξ) in theinterval

[0, n3 − δξ

]; additionally it is always higher than ξ/2 in the interval

[n3 − δξ,+∞

]and

always lower than ξ/2 in the interval [−∞, 0].• Finally, the optimum hedging quantity line converges to ξ/2 while4 → ±∞; yet multi-optimality

emerges simultaneously.

5.1.2 Analysis of Two-Period Model With Changing Values of n3

Another important parameter about the financial situation of the company is n3 in this model. It isan important parameter and may also contain uncertainty in real life cases. Thus, an analysis on theeffect of n3 by simulation is necessary. A typical behavior of the optimum hedging amount functionby changing n3 values is given at Figure 5.8 when the model is used with the base parameters.

37

Figure 5.8: Simulated optimum hedging decisions corresponding to changing values of n3 for two-period model with base parameters

In Figure 5.8, X axis shows continuously changing n3 values while Y axis shows the optimum hedgingdecisions for corresponding n3 values. In both extreme sides of X axis, multi-optimality is observedas suggested in Theorem 7 and the points where multi-optimality started to be observed will also beexamined in this section. On the other hand, the behavior of the optimum hedging amount functionshould be inspected by using Theorem 3 which states:

x∗1,3 =

ξ4/ (24 − n3 + δξ) i f 4 (4 − n3 + δξ) = 0

−ξ4/ (−n3 + δξ) i f 4 (4 − n3 + δξ) 5 0

The expressions ξ4/ (24 − n3 + δξ) and −ξ4/ (−n3 + δξ) will be drawn while n3 is changing and theywill be labeled as Line 1 and Line 2, respectively at Figure 5.9. Figure 5.9 shows the optimum valuessuggested by Theorem 3 with corresponding n3 values and it is observed that these lines indicate theexact solution when two parts of lines 1 and 2 are considered together where 0 5 x∗1,3 5 ξ. However,multi-optimality occurs while n3 goes to ±∞ as Theorem 8 suggests.

The optimum hedging amount x∗1,3 is equal to ξ when 4 = n3 − δξ as stated in Theorem 5. It is satisfiedby n3 = 120 when predefined base values of the parameters are used. As a result, the optimum linepeaks at n3 = 120 and it decreases while getting away from that point. By using Theorem 3, it isobserved that the curves are symmetric with respect to the line n3 = 120. Additionally, the optimumdecision curve is converging to X axis while n3 goes to ±∞.

The behavior of the optimum value function while n3 is changing is similar to δ. However, the magni-tude of the slopes and the point where it peaks is dependent to ξ. Therefore, analysis of the optimumhedging decision with respect to n3 represents the analysis with respect to the corresponding value δas long as the magnitude of ξ stays the same.

38

Figure 5.9: Theoretical optimum hedging decisions corresponding to changing values of 4 for two-period model with base parameters

5.2 Analysis of Three-Period Model

Similar to two-period model, the variables 4 (cash balance of the company at t = 2 without hedging),n3 and n4 are the most important variables.

A set of values of the parameters will be taken into account as a base for numerical analysis. In thisbase parameter value set, the values of the variables are 4 = y1 − n2 = 100, n3 = 30, n4 = 30, ξ = 100,p = 11, λ = 0.5, F1,4 = 10, σ = 1, r = 0.1, β = 0.95. Besides, amounts less then ζ = 0.001 units ofexpected financial distress cost is supposed to be insignificant for the company. Similar to two-periodproblem, multi-optimal decisions exist for the company when difference of financial distress costsare less than ζ. Expected total financial distress costs with base parameter values and with respectto changing values of x1,3 are shown at Figure 5.10. Similar to the numeric analysis for two-periodmodels, for each x1,4 decision, financial distress cost is calculated as the average financial distress costfor 100,000 different realizations of the futures and commodity prices.

X axis denotes different x1,4values while Y axis shows the corresponding expected total financial dis-tress costs in Figure 5.10. As observed from Figure 5.10, the financial distress cost function is convexand the optimum decision for the model with base parameter value set is 44 which is less than ξ.Moreover, at time 1, it is observed that hedging amount gets smaller in expected value as time to goincreases, i.e. 0 5 x∗1,4 5 x∗2,4 5 x∗3,4 = ξ.

5.2.1 Analysis of Three-Period Model With Changing Values of 4

The first term to be analyzed is again 4. Typical behavior of the optimal hedging amount function bychanging 4 values and with base parameter set is obtained, and shown at Figure 5.11.

39

Figure 5.10: Expected total financial distress costs for three-period model with base parameters fordifferent x1,4 values

Figure 5.11: Optimum hedging amounts at t = 1 corresponding to changing values of 4 for three-period model with base parameters

40

Figure 5.12: Optimum hedging decisions for t = 1 and expected optimum hedging decisions for t = 2for three-period model at t = 1 with changing values of y1 and base parameters

Just as Theorem 11 suggested, hedging is becoming a value neutral action while 4 is going through ±∞in Figure 5.11. Therefore, as long as hedging quantity stays in the reasonable limits for corresponding4 values, multi-optimality exists in both sides of Figure 5.11. It is also observed that the optimumhedging decision is equal to 0 when 4 = 0 and 4 = n3 = 30.

As obtained from Theorem 9, the hedging policy will be as 0 5 x∗1,4 5 x∗2,4 5 x∗3,4 = ξ. Therefore, theoptimum hedging decision at time 1 is smaller than or equal to expected optimum hedging decision attime 2 and it is smaller than ξ when the company considers hedging at t = 1. However, the expectedvalue of x∗2,4 at time 1 may be very different at time 2, after the futures prices and cash flows related tofuture contracts are realized. Figure 5.12 shows the optimal hedging decision at t = 1 and the expectedoptimal hedging decision at t = 2 for three-period model with base parameter setting and withn2 = 0,as well as the hedge quantity at t = 1 for two-period model with base parameter set. As observed fromthe figure, optimum hedging quantity at t = 1 is always lower than or equal to the optimum expectedhedging quantity at t = 2.

Similar to two-period model, 4 = y1 − n2 is an effective parameter to the hedging decision and it isalso important for real life decisionmakers. As obtained from Theorem 9, the hedging policy will beas 0 5 x∗1,4 5 x∗2,4 5 x∗3,4 = ξ where expected values are considered for future decisions when thecompany is at time 1.

5.2.1.1 Effect of n3 while 4 values change

It is known that n3 is an important variable and it is one of the most effective factors on hedgingquantity for three-period model. Figure 5.13 shows the respective optimum hedging decisions at time1 while 4 changes and n3 has three different values as 0, 15, 30 while the other parameters are set to

41

Figure 5.13: Optimum hedging decisions corresponding to changing values of 4 and n3 for three-period model with base parameters

their predefined base value levels.

The first thing to notice in Figure 5.13 is that the company hedges nearly to amount of zero when thecash reserve of the company at t=1 except hedging is zero, i.e. 4 = y1 − n2 = 0. This is a similar resultto the two-period model. The reason is that the company will have financial distress at time 2 cost at50% probability if it hedges.

Another important thing to notice is that the optimum hedging quantity becomes zero when the cashreserve of the company at t=3 except hedging is zero, i.e. 4 − n3 = 0. Thus, the company does nothedge at the points where 4 = n3 in Figure 5.13. Another observed point is that the optimum decisionlines are converging to ξ/2 while 4 → ±∞ and this behavior is similar to the optimum solution at time1 for two-period model.

5.3 Analysis of the Approximation of the Model with Compound Interest

In this section, numeric comparison of compound and simple interest model will be performed for two-period model. The base parameter set is the same as the one with simple interest. Typical behavior ofthe optimum decision function as 4 values are changing is obtained when the simulation model withbase parameters is used, and shown at Figure 5.14. Similar to the model with simple interest, hedgingis a value neutral action while 4 → ±∞. Likewise, the optimum decision lines are also converging toξ/2 while 4 → ±∞.

As mentioned in the previous parts, the model with simple interest rates is aimed to be used in esti-mation of the optimum hedging decision at t=1 for the model with compound interest. The discountfactor β is chosen to be 1 to equate the weight of both of financial distress costs in the model withcompound interest, just as simple interest model. The comparison of these two models with the base

42

Figure 5.14: Optimum hedging decisions corresponding to changing values of 4 for two-period modelwith compound interest and base parameters

parameters (except β = 1) is shown in Figure 5.15.

Similar to the model with simple interest, there is a flat line between 4 = (n3 − δξ) / (1 + r) and 4 = 0when n3 − δξ < 0 and β = 1. It is known that the expression (n3 − δξ) is equal to -20 for the baseparameters, which is lower than zero. As an illustration to this case, Figure 5.15 shows the optimumhedging amounts corresponding to changing values of 4 with base parameters. It is observed that,the optimum decision of the model with compound interest is less than the other one as long as 4 isbetween −∞ and −n3 + δξ. Another fact is that, all possible optimum decisions when 4 < 0 are lessthan ξ/ (1 + r).

In case of n3 − δξ > 0, again a flat line starting from the origin takes place in the optimum hedgingdecision figures; yet this line may reach to the level x∗1,3 = ξ in extreme positive values of n3 − δξ.Figure 5.16 shows that two models almost behave the same in the interval (0,∞) of 4 when β = 1 andn3 − δξ = 20, which is more than zero. It is seen that, the optimum hedging quantity of the model withcompound interest is lower than the one with simple interest in the interval (−∞, 0) of 4.

When the expression n3 − δξβ 0 , multi-optimality exists around 4 = 0; and this is also a similarbehavior of the model with simple interest. Besides the optimum hedging quantity becomes closeto ξ/2 while 4 goes to ±∞. Figure 5.17 shows the behavior of the optimum hedging decisions oftwo-period model with compound interest rates when n3 − δξβ = 0. Minimum and maximum valuesof optimum decision sets are shown as separate lines because there is multi-optimality. It may beconcluded that the optimum hedging quantity line for model with compound interest and β = 1 has asimilar shape to the one with simple interest when Figure 5.17 and Figure 5.5 are compared.

Although the models with compound and simple interest rates are pretty much similar when β = 1,

43

Figure 5.15: Optimum hedging decisions of two-period models with simple and compound interestcorresponding to changing values of 4 by using base parameters (except β = 1)

Figure 5.16: Optimum hedging decisions of two-period models with simple and compound interestcorresponding to changing values of 4 by using base parameters (except β = 1 and n3 = 70)

44

Figure 5.17: Optimum hedging decisions of two-period model with compound interest correspondingto changing values of 4 by using base parameters (except β = 1 and n3 = 50)

differences become more apparent while β decreases. Figure 5.18 shows the optimum hedging decisionof the company at time 1 with compound interest rates and for different β and 4 values. It is observedthat, the optimum hedging amount decreases while β decreases. The decisionmakers should alsoconsider β as an important factor for the hedging decision.

In conclusion, the model with simple interest may be helpful in estimating the optimum hedgingamount of the model with compound interest. Two models behaves very similar between max (0, n3 − δξ)and +∞ when β = 1. On the other hand, the optimum hedging decision at time 1 for the model withcompound interest decreases in some regions while β is getting smaller. Briefly, the optimum hedgingamount with compound interest at time 1 is always less than or equal to the optimum hedging quantitywith simple interest.

5.4 Analysis of One-Period Model With Call Option

As stated in the mathematical model, buying call option decision should be examined and profitabilityof this decision with respect to the model with futures contract should be compared. The same baseparameter set is used in these comparisons. In this base setting, the values of the variables are chosenas y1 = 100, n2 = 30, ξ = 100, p = 11.5, λ = 0.5, F1,2 = 10, σ = 1, r = 0.1, β = 0.95 and ζ = 0.0001.By using the value of F1,2 and σ, the value of h is found as approximately 0.4.

The first step for testifying claims of Theorem 15 and 16 is to prove that expected total financial distresscost is convex with respect to x1,2. Possible hedging decisions and corresponding expected financialdistress costs in the model with base settings are shown in Figure 5.19. For each x1,2 decision inFigure 5.19, FDC represents the average total financial distress cost for 100, 000 different realizationsof the future and commodity price realization. As observed, the function is convex with respect to the

45

Figure 5.18: Optimum hedging decisions of two-period model with compound interest correspondingto changing values of 4 and β by using base parameters

decision variable x1,2.

The next step is to examine a typical behavior of the function. Figure 5.20 shows the optimum decisionsets (by denoting highest and lowest optimum values) corresponding to changing values of y1. As

suggested in Theorem 16, y1h +

((p−λ−F1,2)ξ−n2

h

)−is always an optimum decision for the company where

y1h +

((p−λ−F1,2)ξ−n2

h

)−> ξ. It is also observed that the solution y1

h +

((p−λ−F1,2)ξ−n2

h

)−is always an optimum

solution as suggested in our intuition.

Finally a numerical comparison between using futures and call options should be made in order tounderstand which policy is more profitable for the company. It is already known that the optimumhedging decision is always equal to ξ regardless of any other parameters and ξ = 100 in the baseparameter set. On the other hand, the optimum amount of call options are shown at Figure 5.20 forbase parameter set and changing values of y1. Corresponding financial distress costs at optimum policyof two different models for base parameter settings and different y1values are shown in Figure 5.21.

As observed, buying future contract (or forwards) is a better choice in the case of one-period modelsince the firm sets the price of the commodity to a fixed price without paying a considerable amount ofmoney. However, the firm pays the risk premium while buying call options and this causes a decreasein its cash balance. Therefore, expected financial distress cost becomes more when option contract isused.

46

Figure 5.19: Expected total financial distress costs corresponding to different x1,2 values for one-periodmodel with call options and with base parameters

Figure 5.20: Optimum hedging decisions at t = 1 corresponding to changing values of 4 for one-periodmodel with call options and with base parameters

47

Figure 5.21: Expected financial distress costs of one-period models with future and option contractscorresponding to different y1 values at optimum policy

CHAPTER 6

CONCLUSION AND FUTURE WORK

6.1 Conclusion

This study focuses on hedge decision of a company mainly using a specific commodity for a projectwith a predetermined delivery date and amount. Besides it benefits from a subcontractor to producethe necessary output for the project which consists of the commodity. The revenue from the customerof the company is at a fixed amount and is received at the end of the project; however the cost paidto subcontractor is changing with respect to the spot price of the commodity at the payment date andagain is to be paid at the delivery date when revenue is received. Therefore, the company is open to theprice risk and it may use a proper hedge policy to avoid facing financial distress. As well as financialtransactions due to the project, the company has also previously planned payments to be paid duringthe project. The decision is examined both under simple and compound interest rates.

The outcomes of mathematical models and computational results suggests that the firm needs to hedgeat the same amount of commodity needed for the project when using forward contracts or futurecontracts in one-period (it means that margin call requirements of future contracts are completelyignored). By having such an action, the risk of financial distress at the end of the project due tochange in price of the commodity is fully eliminated. On the other hand, the firm needs to decreasehedge amounts while the number of periods is increasing because of another risk; the risk of financialdistress during the project due to margin calls of future contracts. This risk is completely eliminatedwhen hedge amount is zero. Since the risk of financial distress at the end of the project is eliminatedat a perfect hedge policy, the firm needs to have a balance between these two risks; as a result the firmneeds to hedge at an amount between zero and the amount of commodity necessary for the project.Besides, the optimum hedge amount decreases in average while the number of periods where margincalls are adjusted are increasing, because the risk of financial distress due to margin call requirementsis increasing.

Decision of buying call options, with a strike price of the expected commodity price at the end of theproject, is also examined in one-period framework in order to benchmark the profitability with thehedge decision via futures. The optimum hedging policy via call options is mainly analyzed by usingnumerical results. When using call options, it is found out that the firm may both under-hedge or over-hedge in expected value maximization perspective with respect to the values of the parameters in themodel. In addition, according to the numerical results of our models, hedging via futures (or forwardcontracts) is more profitable than hedging via call options in one-period.

49

6.2 Future Research

In this study, decision of hedging via futures contract of a company under price risk is elaboratelyanalyzed in different number of periods of time and through expected value maximization framework.However, initial margins of future contracts are not taken into account in these studies. Therefore, thisstudy can be improved by considering different initial margin models.

Call option decision with a strike price of the expected price of the commodity in the end of the project(which is equal to the future contract price of the commodity maturing at the end of the project) is alsoexamined in this study. However, further analyzes are needed in order to fully analyze the decision ofhedging via option contracts. The decision should be analyzed in different time frames and especiallywith different strike price levels.

50

REFERENCES

Basak, S. and Chabakauri, G., "Dynamic hedging in incomplete markets: a simple solution" Reviewof financial studies (2012), 25(6), 1845-1896,

Cecchetti, S. G., Cumby, R. E. and Figlewski, S., “Estimation of the Optimal Futures Hedge” TheReview of Economics and Statistics Vol. 70, No. 4 (November, 1988), pp. 623-630

Dau, C. and Groch, M., "Approaches for Avoiding Margin Call Situations Through Risk-ManagementAutomation." (2005)

Garner, C., “Adjusting Margin and Risk : Tips and Tricks to Reduce Margin Requirements and Alle-viate Margin Calls” Upper Saddle River, N.J. : FTPress Delivers, (2010)

Goel, A. and Gutierrez, G.J., “Integrating Commodity Markets in the Optimal Procurement Policiesof a Stochastic Inventory System” , University of Texas-Austin (2009)

Goel, A. and Gutierrez, G.J., “Multiechelon Procurement and Distribution Policies for Traded Com-modities” Management Science, Vol. 57, No. 12, (December 2011), pp. 2228-2244

Heifner, R. G., “Optimal Hedging Levels and Hedging Effectiveness Cattle Feeding” AgriculturalEconomics Research 24 (April 1972), pp. 25-36

Judge, A., “Why and How UK Firms Hedge” European Financial Management (June 2006) v. 12, iss.3, pp. 407-41

Maes, S.P., “Integrated Operational and Financial Risk Management: a Flour Milling Case”, Master ofScience in Operations Management and Logistics, Eindhoven University of Technology, (2011), 75p

McDonald, R. L., Cassano, M., and Fahlenbrach, R., "Derivatives markets" Addison-Wesley, (2006)

Modigliani, F. and M.H. Miller, “The Cost of Capital, Corporation Finance, and the Theory of Invest-ment” American Economic Review 48, (1958), pp. 261-297

Rolfo, J., “Optimal Hedging under Price and Quantity Uncertainty: The Case of a Cocoa Producer”Journal of Political Economy 88 (February 1980), pp. 100-116

Smith, C. and R. Stulz, “The Determinants of Firms’ Hedging Policies” Journal of Financial andQuantitative Analysis 20, (1985), pp. 391-405

Stulz, R. M., "Risk management and derivatives" South-Western Publishing, (2003)

Tanrisever, F. and Gutierrez, G.J., “An Integrated Approach to Commodity Risk Management” Work-ing paper, University of Texas-Austin (2011)

Tanrisever, F. and Levij, L., “Electricity Trading Risk Management” (2011), Working paper, Technis-che Universiteit Eindhoven

Tanrisever, F., Duran, S. and Sumer, P., “Integrated Risk Management Applications in Offshore Wind-farm Construction” Master of Science in Industrial Engineering, Middle East Technical University,(2011)

Winkler, R. L., Roodman, G. M. and Britney, R. R., “The Determination of Partial Moments”, Man-agement Science, Vol. 19, No. 3, (November, 1972), pp. 290-296

52

APPENDIX A

LIST OF NOTATIONS

In this study, discrete points in time are used and time i happens before time i+1. The random variablesare shown with the sign “∼” on the top of the variable whereas the realization of the correspondingrandom variables are denoted as standard uppercase letters.

S i: Unit price of the commodity at the time i

p : Unit contract price fixed at time 1 to be paid by the customer to the company

(S i + λ): Unit price paid by the company to the subcontractor for asset delivery at time i.

ξ : Units of commodity required for the project

Fi, j:Futures contract price of commodity starting at time i, maturing at time j

xi, j : Amount of futures or call options longed at time i for the maturity time j

ni : Cash outflow of the company at time i

yi : Cash reserve of the company at time i after the transactions are realized except FDC premium

rd : Minimum required rate of return which is used to discount future cash flows and is assumed to beconstant for all periods

β : The multiplier used to discount a cash flow to the previous time point, formulized as 1/ (1 + rd)

r : Cost of financial distress per unit which is assumed to be constant at all time points

φQJ (.) : Risk neutral probability density function of random variable J

EQJ (.) : The expected value function which is found by using risk neutral probability function of

random variable J

4Vi+1 (yi) : Expected future cash change of the company discounted to respective value of time i + 1when the company is at time i

Vi+1 (yi) : Maximum value of the expected future cash change of the company discounted to respectivevalue of time i + 1 when the company is at time i , i.e. Vi+1 (yi) = max4Vi+1 (yi)

4 : Net cash balance of the company after the preplanned cash transactions are completed at time 2,i.e. 4 = y1 − n2

δ : Expected unit profit margin of the company, calculated as p − λ − S i for a project starting at time 1

53

and ending at time i

h : Unit risk premium price of option contract

54

APPENDIX B

PROOFS

B.1 Theorem 1

The first derivative of the term (−4V2 (y1)) with respect to x1,2 is shown as the following:

d (−4V2 (y1))dx1,2

=d∫ ∞

0

[n2 −

(p − S 2 − λ

)ξ −

(S 2 − F1,2

)x1,2 + βr

[y2

]−] φQs2 (S 2) dS 2

dx1,2

=d∫ ∞

0

[(F1,2 − S 2

)x1,2 + βr

[y2

]−] φQS 2

(S 2) dS 2

dx1,2

=d∫ ∞

0 βr[y2

]− φQS 2

(S 2) dS 2

dx1,2

Since x1,2 is included only in the terms(F1,2 − S 2

)x1,2 and −r

[y2

]−, the derivatives of these terms aretaken. However, it is already known that∫ ∞

0 S 2φQS 2

dS 2 = F1,2. Consequently, the only term left over is the term referring financial distress cost,i.e. r

[y2

]−. As a result, the expression can be transformed into the following by writing the term r[y2

]−explicitly:

d (−4V2 (y1))dx1,2

=d∫ ∞

0βr

[y1 +

(S 2 − F1,2

)x1,2 +

(p − S 2 − λ

)ξ − n2

]−φQ

S 2(S 2) dS 2

dx1,2

For y2 to be negative, there are two cases to be considered. The cases are shown as the following where−y1+F1,2 x1,2−(p−λ)ξ+n2

x1,2−ξis denoted as a:

y2 < 0⇔

S 2 < a i f x1,2 > ξ

S 2 > a i f x1,2 < ξ

By using the only the terms having x1,2 as a multiplier and Leibniz Rule on taking derivative of y2, thefirst derivative of the function (−4V2 (y1)) can be referred as the following:

d (−4V2 (y1))dx1,2

=

βr∫ a

0

(F1,2 − S 2

)φQ

S 2(S 2) dS 2 f or x1,2 > ξ

βr∫ ∞

a

(F1,2 − S 2

)φQ

S 2(S 2) dS 2 f or x1,2 < ξ

The point where the first derivative of the function is zero will be an extreme point. The main termin both of the cases are partial moments of normal distribution of order 1. Therefore, for the case of

55

x1,2 > ξ, the first derivative of V2 (y1) is positive and it is vice versa for the case of x1,2 < ξ. By referringto the paper “the Determination of Partial Moments” written by Winkler, Roodman and Britney (1972),the first derivative of the function can only be equal to zero when x1,2 = ξ.

At the next step, convexity of the term (−4V2 (y1)) should be examined. In the term (−4V2 (y1)), theexpression n2 −

(p − S 2 − λ

)ξ −

(S 2 − F1,2

)x1,2 is linear in x1,2. Therefore the term βr

[y2

]− should beexamined. Obviously the term is a piecewise linear function in x1,2 too. It is known that if a functionis convex, its expected value function is also convex. Finally the term (−4V2 (y1)) is concluded to beconvex, because all parts of it are convex. Due to convexity, the extreme point found is the universalminimum point of the function. Therefore, the decision x∗1,2 = ξ is proved to be the optimum hedgequantity.

B.2 Theorem 2

In order to prove the theorem, firstly the term of (−4V2 (y1)) should be minimized similar to the proofof Theorem 1. The term (−4V2 (y1)) at t = 1 can be described mathematically as the following afterreplacing x2,3 by ξ due to Lemma 1:

(−4V2 (y1)) =

∞∫0

[n2 −

(F2,3 − F1,3

)x1,3

+ β

∞∫0

[n3 −

(p − S 3 − λ

)ξ −

(S 3 − F2,3

)ξ

+ β(r[y2

]−+ r

[y3

]−)]φQS 3

(S 3) dS 3

]φQ

F2,3

(F2,3

)dF2,3

wherey2 = y1 − n2 +

(F2,3 − F1,3

)x1,3

y3 = y2 − n3 +(p − F2,3 − λ

)ξ

In order to find the optimum hedging quantity at time 1, it is necessary to analyze the first and thesecond derivatives of the term (−4V2 (y1)) with respect to the decision variable at t = 1, i.e. x1,3. SinceF1,3 =

∫ ∞0

[(F2,3

)]φQ

F2,3dF2,3 and the rest of the expression do not contain x1,3 except financial distress

cost premiums, the first derivative of the term (−4V2 (y1)) is the following.

d (−4V2 (y1))dx1,3

=

β d(∫ ∞

0

[r[y2

]−+

∫ ∞0 r

[y3

]− φQS 3

dS 3

]φQ

F2,3

(F2,3

)dF2,3

)dx1,3

In order to examine the expression above, the breakpoints where y2 and y3 becomes zero should befound. The condition that makes the expression y2 = y1 −n2 +

(F2,3 − F1,3

)x1,3 becomes less than zero

is the following where b =−y1+F1,3 x1,3+n2

x1,3.

56

y2 < 0⇔

F2,3 < b i f x1,3 > 0

F2,3 > b i f x1,3 < 0

Since the expected financial distress cost at the end of the second period is also aimed to be found, theconditions that satisfy y3 < 0 should be examined. Thus y3 will be equal toy1+

(F2,3 − F1,3

)x1,3+

(p − F2,3 − λ

)ξ−n3−n2. The condition for y3 < 0 where c =

−y1+F1,3 x1,3−(p−λ)ξ+n2+n3

x1,3−ξ

is the following.

y3 < 0⇔

F2,3 < c i f x1,3 > ξ

F2,3 > c i f x1,3 < ξ

We will prove that the firm should underhedge in order to decrease the total of the expected financialdistress costs. By examining the negativity conditions of y2 and y3, and corresponding first derivativeof the objective function in these 2 different intervals:

Case I: x1,3 > ξ

As a result, the expression will be as the following.

d (−4V2 (y1))dx1,3

=

d[βr

∫ b0

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

]dx1,3

+d[βr

∫ c0

[−y1 +

(F1,3 − F2,3

)x1,3 + n2 −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

]dx1,3

This expression can be transformed into the following by only considering the terms including x1,3 anduse of Leibniz rule:

d (−4V2 (y1))dx1,3

= βr

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

c∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

Both of the terms βr∫ b

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 and

βr∫ c

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 are positive. Consequently, there cannot be an extreme point when

x1,3 > ξ.

Case II: 0 5 x1,3 5 ξ

In this case we have:

57

d (−4V2 (y1))dx1,3

=

d[βr

∫ b0

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

]dx1,3

+dβr

∫ ∞c

[−y1 +

(F1,3 − F2,3

)x1,3 + n2 −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

This expression can be transformed into the following to use Leibniz rule easier.

d (−4V2 (y1))dx1,3

= βr

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

∞∫c

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

The term βr∫ b

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 is positive and the term

βr∫ c

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 is negative. Consequently, an extreme point may exist when 0 5

x1,3 5 ξ.

Case III: x1,3 < 0

In this case the company is selling futures contracts. As a result, the expression will be as the following.

d (−4V2 (y1))dx1,3

=d[βr

∫ ∞b

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

]dx1,3

+dβr

∫ ∞c

[−y1 +

(F1,3 − F2,3

)x1,3 + n2 −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

Ignoring terms not including x1,3 and use of Leibniz rule, we have:

d (−4V2 (y1))dx1,3

= βr

∞∫b

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

∞∫c

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

Both of the terms are negative. Consequently, an extreme point may not exist when x1,3 < 0.

Similar to Theorem 1, the all terms except β(r[y2

]−+ r

[y3

]−) are linear in x1,3 in −4V2 (y1). Besides,it is known that both of the financial distress costs at time 2 and 3 are piecewise linear functions in x1,3.It is known that if a function is convex, its expected value function is also convex. Finally the term

58

−4V2 (y1) is concluded to be convex, because all parts of it are convex. Due to convexity, the extremepoints found are universal minimum points of the function. In other words, they are the solutions thatgenerate the minimum value of the term. Consequently, there cannot be an optimum solution whenx1,3 > ξ or x1,3 < 0, optimum solution can take place when 0 5 x∗1,3 5 ξ.

B.3 Theorem 3

We can rewrite b and c as(F1,3 −

4

x1,3

)and

(F1,3 +

−4+n3−δξx1,3−ξ

)respectively. Setting d(−4V2(y1))

dx1,3= 0 at case

II (which suggests 0 5 x1,3 5 ξ) at Theorem 2, we get∫ F1,3−

4x1,3

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 =∫ F1,3+

−4+n3−δξx1,3−ξ

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 since the random variable F2,3 has a normal distribution with

mean F1,3. Indeed, in order to satisfyd

dx1,3(−V2 (y1)) = 0, the following equation should also be satisfied:

∣∣∣∣∣∣ 4x1,3

∣∣∣∣∣∣ =

∣∣∣∣∣∣4 − n3 + δξ

ξ − x1,3

∣∣∣∣∣∣Case I: 4 (4 − n3 + δξ) > 0

Suppose that 4 > 0 and 4 − n3 + δξ > 0, or 4 < 0 and 4 − n3 + δξ < 0.

ξ4 − x1,34 = x1,34 − n3x1,3 + δξx1,3

Finally x∗1,3 = ξ4/ (24 − n3 + δξ) satisfying x∗1,3 < ξ and 4 (4 − n3 + δξ) > 0.

Case II: 4 (4 − n3 + δξ) < 0

Suppose that 4 < 0 and 4 − n3 + δξ > 0, or 4 > 0 and 4 − n3 + δξ < 0.

−ξ4 + x1,34 = x1,34 − n3x1,3 + δξx1,3

Finally x∗1,3 = −ξ4/ (−n3 + δξ) satisfying x∗1,3 < ξ and 4 (4 − n3 + δξ) > 0.

B.4 Theorem 4

When 4 = y1 − n2 = 0, we have b = F1,3 and c = F1,3 +δξ−n3ξ−x1,3

. Thus, the first derivative of the term(−4V2 (y1)) in the region where optimal hedging amount is found 0 5 x1,3 5 ξ is:

d (−4V2 (y1))dx1,3

= βr

F1,3∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

∞∫F1,3+

δξ−n3ξ−x1,3

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

59

It is already known that∫ F1,3

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+∫ ∞

F1,3+δξ−n3x1,3−ξ

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 > 0, as a result d

dx1,3(−4V2 (y1)) is positive. Indeed, increas-

ing x1,3 will increase the expected financial distress cost and we already know that the optimum deci-sion should satisfy 0 5 x1,3 5 ξ. Consequently, x∗1,3 is equal to 0 when 4 = 0.

B.5 Theorem 5

It is already known that b = F1,3 −4

x1,3and c = F1,3 +

−4+n3−δξx1,3−ξ

. When 4 = n3 − δξ, we have c = F1,3.Finally the first derivative of (−4V2 (y1)) is the following for x1,3 < ξ:

d (−4V2 (y1))dx1,3

= βr

F1,3−4

x1,3∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

∞∫F1,3

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

Since∫ F1,3−

4x1,3

0

(F1,3 − F2,3

)φQ

F2,3dF2,3 +

∫ ∞F1,3

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 < 0, the expression above is

negative and the decisionmaker should increase its hedging quantity until it reaches to ξ. The optimumhedging decision is full-hedging, i.e. x∗1,3 = ξ

B.6 Theorem 6

When 4 = y1−n2 = n3−δξ = 0, we have b = c = F1,3, and the first derivative of the objective functionis for 0 5 x1,3 5 ξ:

d (−4V2 (y1))dx1,3

= βr

F1,3∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

∞∫F1,3

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

= 0

In this case, hedging quantity is not important in terms of financial distress cost minimization.

Figure B.1 shows numerical simulation results of a model with 100, 000 runs and values n3 = 50,ξ = 100, p = 11, λ = 0.5, F1,3 = 10, σ = 1, r = 0.1, β = 0.95. Besides, amounts less then ζ = 0.0001units of expected financial distress cost is supposed to be neglected for the company.

60

Figure B.1: Expected total financial distress costs by changing values of x1,3 for two-period modelwith base parameters

In Figure B.1, two different lines show the maximum and minimum hedging amounts for the decision-maker at optimality. Moreover, the dashed line shows the numerically calculated optimum points. Asobserved from Figure B.1, multi-optimal decisions exist for the company when getting closer to thepoint where 4 and n3 − δξ equal to 0.

B.7 Theorem 7

As stated above, the optimum hedging policy follows the function below.

x∗1,3 =

ξ4/ (24 − n3 + δξ) i f 4 (4 − n3 + δξ) = 0

−ξ4/ (−n3 + δξ) i f 4 (4 − n3 + δξ) 5 0

The optimum hedging decision line will follow −ξ4/ (−n3 + δξ) when min (0, n3 − δξ) < 4 <max (0, n3 − δξ). On the other hand, at the both edges while 4 goes to ±∞, the optimum hedgingdecision will be equal to ξ4/ (24 − n3 + δξ). If the limits of the lines are calculated, it can already beseen that:

lim4→±∞

(ξ4/ (24 − n3 + δξ)) = ξ/2

It means that the optimum hedging quantity is ξ/2 while 4 goes to ±∞. However it should also benoted that the normal distribution is used in the model. Thus, the total expected financial distress costdoes not change in significant amounts in both infinite ends and the behavior of the function should beexamined in the region where we have the optimal hedging decisions 0 5 x∗1,3 5 ξ, and 4 → ±∞. As4 → ∞; b → −∞ and c → ∞; and as 4 → −∞; b → ∞ and c → −∞. Finally, the first derivative of(−4V2 (y1)) is:

61

d (−4V2 (y1))dx1,3

= βr

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

∞∫c

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

= 0

The expressions∫ ∞

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 and

∫ ∞∞

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 are known to

be zero. As a result, there is no effect of hedging in these conditions as shown above.

Figure B.2 shows numerical simulation results of a model with 100, 000 runs and values n3 = 30,ξ = 100, p = 11, λ = 0.5, F1,3 = 10, σ = 1, r = 0.1, β = 0.95. Besides, amounts less then ζ = 0.0001units of expected financial distress cost is supposed to be neglected for the company.

Figure B.2: Expected total financial distress costs by changing values of x1,3 for two-period modelwith base parameters

In Figure B.2, two different lines show the maximum and minimum hedging amounts for the decision-maker at optimality. Moreover, the dashed line shows the numerically calculated optimum points. Asobserved from Figure B.2, multi-optimal decisions exist for the company when 4 → ±∞. However,the calculated optimum quantity line converges to ξ/2.

B.8 Theorem 8

As stated above, the optimum hedging policy follows the function below.

x∗1,3 =

ξ4/ (24 − n3 + δξ) i f 4 (4 − n3 + δξ) = 0

−ξ4/ (−n3 + δξ) i f 4 (4 − n3 + δξ) 5 0

62

The optimum hedging decision line may follow both of the scenarios with respect to the sign of 4 andn3 while n3 goes to ±∞. When 4 > 0 and n3 goes to +∞ or 4 < 0 and n3 goes to −∞, the optimumhedging decision at t = 1 will follow the line −ξ4/ (−n3 + δξ). On the contrast, x∗1,3 will follow the lineξ4/ (24 − n3 + δξ) when 4 < 0 and n3 goes to +∞ or 4 > 0 and n3 goes to −∞. Thus the optimumdecision will converge to 0 in both ends as shown below.

lim4→±∞

(ξ4/ (24 − n3 + δξ)) = 0

lim4→±∞

(−ξ4/ (−n3 + δξ)) = 0

It means that the optimum hedging quantity is 0 while n3 goes to ±∞. However, again it should benoted that the normal distribution is used in the model. Thus, the total expected financial distresscost does not change in significant amounts in both ends and the behavior of the function should beexamined in the region where we have the optimal hedging decisions 0 5 x∗1,3 5 ξ, and n3 → ±∞. Asn3 → +∞ and 4 > 0; x∗1,3 = −ξ4/ (−n3 + δξ), b = F1,2 −

n3+δξξ

and c = F1,2 +n3+δξξ

; b → −∞ and

c → +∞. As n3 → −∞ and 4 < 0; x∗1,3 = −ξ4/ (−n3 + δξ), b = F1,2 −n3+δξξ

and c = F1,2 +n3+δξξ

;

b → +∞ and c → −∞. As n3 → +∞ and 4 < 0; x∗1,3 = ξ4/ (24 − n3 + δξ), b = F1,2 −24−n3+δξ

ξand

c = F1,2 +24−n3+δξ

ξ; b → +∞ and c → −∞. As n3 → −∞ and 4 > 0; x∗1,3 = ξ4/ (24 − n3 + δξ),

b = F1,2 −24−n3+δξ

ξand c = F1,2 +

24−n3+δξξ

; b → −∞ and c → +∞. Consequently, the first derivativeof (−4V2 (y1)) is known to be the following expression:

ddx1,3

(−4V2 (y1)) = βr

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

∞∫c

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

= 0

Since, the expressions∫ 0

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3,∫ ∞

∞

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 and

∫ ∞0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 are all known to be equal to

zero. Therefore, in the region 0 5 x∗1,3 5 ξ that we considered, there is multioptimality due tocharacteristics of normal distribution.

Figure B.3 shows numerical simulation results of a model with 100, 000 runs and values y1 = 100,n2 = 30, n3 = 30, ξ = 100, p = 11, λ = 0.5, F1,3 = 10, σ = 1, r = 0.1, β = 0.95. Besides, amounts lessthen ζ = 0.0001 units of expected financial distress cost is supposed to be neglected for the company.

63

Figure B.3: Expected total financial distress costs by changing values of x1,3 for two-period modelwith base parameters

In Figure B.3, two different lines show the maximum and minimum hedging amounts for the decision-maker at optimality. Moreover, the dashed line shows the numerically calculated optimum points. Asobserved from Figure B.3, multi-optimal decisions exist for the company when n3 → ±∞. However,the calculated optimum quantity line converges to 0.

B.9 Theorem 9

The first derivative is to be examined in order to observe the behavior of the function.

d (−4V2 (y1))dx1,4

=dβ3r

[Eφ

F

([y2

]−+

[y3

]−+ r

[y4

]−)]dx1,4

The expression above should better be analyzed in three different components. First of all, the break-points of financial distress costs in all those components should be found.

It is already known that y2 = y1 +(F2,4 − F1,4

)x1,4 − n2. It behaves different in two different cases

with respect to the value of x1,4. Suppose that e =−y1+F1,4 x1,4+n2

x1,4. Then the negativity condition of y2 is

F2,4 < e when x1,4 > 0; and the condition in case of x1,4 < 0 is F2,4 > e.

• Suppose that x1,4 > 0:

Then the expected value of r[y2

]− will be as the following:

EφF

([y2

]−)=

e∫0

(−y1 +

(F1,4 − F2,4

)x1,4 + n2

)φQ

F2,4

(F2,4

)dF2,4

64

Finally, first derivative of expected value of r[y2

]− is:

d(Eφ

F

([y2

]−))dx1,4

=

e∫0

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4

• Suppose that x1,4 < 0:

Then the expected value of r[y2

]− will be as the following:

EφF

([y2

]−)=

+∞∫e

(−y1 +

(F1,4 − F2,4

)x1,4 + n2

)φQ

F2,4

(F2,4

)dF2,4

First derivative of expected value of r[y2

]−when x1,4 > 0:

d(Eφ

F

([y2

]−))dx1,4

=

+∞∫e

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4

We have y3 = y1 +(F2,4 − F1,4

)x1,4 − n2 +

(F3,4 − F2,4

)x2,4 − n3. Negativity condition of y3 should be

found by assuming that y3 < 0, then the condition becomes the following:

F2,4(x1,4 − x2,4

)< −y1 + F1,4x1,4 + n2 + n3 − F3,4x2,4

For the sake of notation ease, suppose that g = −y1 + F1,4x1,4 + n2 + n3 and f = x1,4 − x2,4. There aretwo cases with respect to value of x1,4 as the following. In case of x1,4 > x2,4, the condition of y3 isF2,4 <

gf . On the other hand, the condition evolves into F2,4 >

gf in case of x1,4 < x2,4.

Note that, the decision variable is x1,4 at t = 1. Therefore, the first derivative of[y3

]− with respect tox1,4 is to be analyzed. Moreover, F1,4 is known, while F2,4 and F3,4 are random stochastic variables att = 1; as a result recursive integral will be used to calculate the expected financial distress cost.

• Suppose that x1,4 > x2,4:

First derivative of r[y3

]− when x1,4 > x2,4:

d(Eφ

F

([y3

]−))dx1,4

=d∫ +∞

F3,4=0

∫ g−F3,4 x2,4f

F2,4=0 (−y3) φQF2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

dx1,4

=

+∞∫F3,4=0

g−F3,4 x2,4f∫

F2,4=0

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

• Suppose that x1,4 < x2,4:

First derivative of r[y3

]− when x1,4 < x2,4:

65

d(Eφ

F

([y3

]−))dx1,4

=

d∫ +∞

F3,4=0

∫ +∞

F2,4=g−F3,4 x2,4

f(−y3) φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

dx1,4

=

+∞∫F3,4=0

+∞∫F2,4=

g−F3,4 x2,4f

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

The term y4 is equal to y1 +(F2,4 − F1,4

)x1,4 − n2 +

(F3,4 − F2,4

)x2,4 − n3 +

(S 4 − F3,4

)x3,4

+(p − S 4 − λ

)ξ − n4. Negativity condition of y4 should be found by assuming that y4 < 0. By using

Lemma 2, it is already known that: x∗3,4 = ξ. Furthermore, Theorem 2 suggests that(x2,4 − ξ

)5 0

regardless of the value of the other variables. Therefore the expression becomes the following:

F2,4(x1,4 − x2,4

)< −y1 + F1,4x1,4 + n2 + n3 + n4 − (p − λ) ξ − F3,4

(x2,4 − ξ

)There are two different conditions for the different cases with respect to the value of x∗1,4. For thesake of notation ease, suppose that f = x1,4 − x2,4, k = −y1 + F1,4x1,4 + n2 + n3 + n4 − (p − λ) ξ andl = x2,4−ξ. The negativity condition of y4 for x1,4 > x2,4 is F2,4 <

k−F3,4lf ; while the negativity condition

for x1,4 < x2,4 is F2,4 >k−F3,4l

f .

• Suppose that x1,4 > x2,4:

The first derivative of[y4

]− with respect to x1,4 is shown below.

d(Eφ

F

([y4

]−))dx1,4

=

d(∫ +∞

F3,4=0

∫ k−F3,4 l1f

F2,4=0 (−y4) φQF2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

)dx1,4

=

+∞∫F3,4=0

k−F3,4 lf∫

F2,4=0

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

• Suppose that x1,4 < x2,4:

The first derivative of[y4

]− with respect to x1,4 is shown below.

d(Eφ

F

([y4

]−))dx1,4

=

d(∫ +∞

F3,4=0

∫ +∞

F2,4=k−F3,4 l

f(−y4) φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

)dx1,4

=

+∞∫F3,4=0

+∞∫F2,4=

k−F3,4 lf

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

The convexity of the function may be proved by using the same methodology used in Theorem 2.Similar to the previous models, it is observed that the expressions

[y2

]− ,[y3]−and

[y4

]−are piecewise

66

convex functions and other variables in −4V2 (y1) are all linear, and if a function is convex, its expectedvalue function is also convex. Consequently, it is obvious that the term −4V2 (y1) is convex in x1,4.

From the point of decisionmaker, there are two different intervals of x1,4.

Case I: x1,4 > x2,4 and x1,4 > 0

d (−4V2 (y1))dx1,4

=

e∫F2,4=0

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4

+

+∞∫F3,4=0

g−F3,4 x2,4f∫

F2,4=0

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

+

+∞∫F3,4=0

k−F3,4 lf∫

F2,4=0

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

This expression is positive, it means that the expected total financial distress cost will increase whilex1,4 is increasing. Indeed, an extreme point cannot exist here; as a result it is not the optimum interval.

Case II: x1,4 < x2,4 and x1,4 > 0

d (−4V2 (y1))dx1,4

=

e∫F2,4=0

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4

+

+∞∫F3,4=0

+∞∫F2,4=

g−F3,4 x2,4f

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

+

+∞∫F3,4=0

+∞∫F2,4=

k−F3,4 lf

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

This expression can either be positive or negative or zero. Therefore, the optimum decision may existin this interval.

Case III: x1,4 > x2,4 and x1,4 < 0

This case suggests that x2,4 < 0. However, this assumption contradicts with our previously provedknowledge via Lemma 2; x2,4 ≥ 0. As a result, there is no need to examine this situation.

Case IV: x1,4 < x2,4 and x1,4 < 0

d (−4V2 (y1))dx1,4

=

∞∫F2,4=e

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4

67

+

+∞∫F3,4=0

+∞∫F2,4=

g−F3,4 x2,4f

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

+

+∞∫F3,4=0

+∞∫F2,4=

k−F3,4 lf

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

This expression can be only negative and it means that the total expected financial distress cost alwaysincreases while x1,4 is increasing in the interval (−∞, 0]. Consequently, there cannot be an optimumdecision in this interval and optimum x∗1,4 should be equal to or higher than 0 because of the convexity.

After examining all the possible cases with respect to the values of the decision variables and convexityof −4V2 (y1), that suggests the optimum decision policy is under-hedging, such that x∗1,4 5 x∗2,4 5 ξ

where x∗2,4 is considered as the mean of the expected optimum hedge amount at time 2.

B.10 Theorem 10

Initially, the mathematical model should be composed. The model of n period model is shown below.

V2 (y1) = minx1,n+1

EφF2,n+1

[n2 +

(F1,n+1 − F2,n+1

)x1,n+1 − βV3(y2)

]Subject to:

y2 = y1 − n2 +(F2,n+1 − F1,n+1

)x1,n+1

V3 (y2) = minx2,n+1

EφF2,n

[n2 +

(F2,n+1 − F3,n+1

)x1,n+1 − βV4(y3)

]Subject to:

y3 = y2 − n3 +(F3,n+1 − F2,n+1

)x1,n+1

.............

Vn (yn−1) = minxn−1,n+1

EφF2,n+1

[nn−1 +

(Fn−1,n+1 − Fn,n+1

)x1,n+1 − βVn+1(yn)

]Subject to:

yn = yn−1 − nn +(F3,n − F2,n

)x1,n

Vn+1(yn) = minxn,n+1

Eφsn+1

[nn+1 −

(p − S n+1 − λ

)ξ +

(Fn,n+1 − S n+1

)xn,n+1 − βVn+2(yn+1)

]

Subject to:yn+1 = yn − nn+1 +

(p − S n+1 − λ

)ξ +

(S n+1 − Fn,n+1

)xn,n+1

There is no decision variable at t = n + 1, as a result the function Vn+2(yn+1) is as the following:

68

Vn+2(yn+1) = −∑n+1

i=2 r[yi]−

Our aim is to solve the problem at t = 1. Thus, the first derivative is to be examined in order to observethe behavior of the function.

d (−4V2 (y1))dx1,n+1

=d[βnr

∑n+1i=2

[yi]−]

dx1,n+1

Similar to the proof of Theorem 9, the expression above should better to be analyzed in three differentcomponents. Initially, the breakpoints of financial distress costs in all those components should befound.

It is already known that y2 = y1 +(F2,n+1 − F1,n+1

)x1,n+1 − n2. Negativity condition of y2 should be

found by assuming that y2 < 0. It behaves different in two different cases with respect to the valueof x1,n+1. Suppose that m1 =

−y1+F1,n+1 x1,n+1+n2

x1,n+1. Then the negativity condition is the following when

x1,4 > 0:

F2,n+1 < m1

The condition in case of x1,n+1 < 0 is the following:

F2,n+1 > m1

• Suppose that x1,n+1 > 0:

Then the expected value of r[y2

]− will be as the following:

EφF

([y2

]−)=

m1∫0

(−y1 +

(F1,n+1 − F2,n+1

)x1,n+1 + n2

)φQ

F2,n+1

(F2,n+1

)dF2,n+1

Finally, first derivative of expected value of r[y2

]− is:

d(Eφ

F

([y2

]−))dx1,n+1

=

m1∫0

(F1,n+1 − F2,n+1

)φQ

F2,n+1

(F2,n+1

)dF2,n+1

• Suppose that x1,n+1 < 0:

Then the expected value of r[y2

]− will be as the following:

EφF

([y2

]−)=

+∞∫m1

(−y1 +

(F1,n+1 − F2,n+1

)x1,4 + n2

)φQ

F2,n+1

(F2,n+1

)dF2,n+1

First derivative of expected value of r[y2

]−when x1,4 > 0:

69

d(Eφ

F

([y2

]−))dx1,n+1

=

+∞∫m1

(F1,n+1 − F2,n+1

)φQ

F2,n+1

(F2,n+1

)dF2,n+1

It is already known that y3 = y1 +(F2,n+1 − F1,n+1

)x1,n+1 − n2

+(F3,n+1 − F2,n+1

)x2,n+1 − n3. Negativity condition of y3 should be found by assuming that y3 < 0,

then the condition becomes the following:

F2,n+1(x1,n+1 − x2,n+1

)< −y1 + F1,n+1x1,n+1 + n2 + n3 − F3,n+1x2,n+1

For the sake of notation ease, suppose thatm2 =

−y1+F1,n+1 x1,n+1+n2+n3−F3,n+1 x2,n+1

(x1,n+1−x2,n+1) . There are two cases with respect to value of x1,4 as the following.In case ofx1,4 > x2,4, the condition is:

F2,n+1 < m2

On the other hand, the condition evolves into the following expression in case of x1,4 < x2,4.

F2,n+1 > m2

Note that, the decision variable is x1,4 at t = 1. Therefore the first derivative of[y3

]− with respectto x1,n+1 is to be analyzed. Moreover, F1,n+1is known, while F2,n+1 and F3,n+1 are random stochasticvariables at t = 1; as a result recursive integral will be used to calculate the expected financial distresscost.

• Suppose that x1,n+1 > x2,n+1:

First derivative of r[y3

]− when x1,n+1 > x2,n+1:

d(Eφ

F

([y3

]−))dx1,n+1

=d∫ +∞

F3,n+1=0

∫ m2

F2,n+1=0 (−y3) φQF2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1

dx1,n+1

=

+∞∫F3,4=0

m2∫F2,4=0

(F1,n+1 − F2,n+1

)φQ

F2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1

• Suppose that x1,n+1 < x2,n+1:

First derivative of r[y3

]− when x1,n+1 < x2,n+1:

d(Eφ

F

([y3

]−))dx1,n+1

=d∫ +∞

F3,n+1=0

∫ +∞

F2,n+1=m2(−y3) φQ

F2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1

dx1,n+1

=

+∞∫F3,n+1=0

+∞∫F2,n+1=m2

(F1,n+1 − F2,n+1

)φQ

F2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1

70

After all those observations, a generalized examination is necessary to produce general formulas. Sup-pose that 3 < i < n + 1. Then yi will be defined as the following:

yi = y3 +∑i

j=4

((F j,n+1 − F j−1,n+1

)x j−1,n+1 − n j

)= y1 +

(F2,n+1 − F1,n+1

)x1,n+1 +

(F3,n+1 − F2,n+1

)x2,n+1

−n2 − n3 +∑i

j=4

((F j,n+1 − F j−1,n+1

)x j−1,n+1 − n j

)For the sake of ease of notation Γ is defined such as Γ = −y1 + F1,n+1x1,n+1 − F3,n+1x2,n+1 + n2 + n3. Thebreakpoint of negativity of yi is also defined as the following:

mi−1 =Γ −

∑ij=4

((F j,n+1 − F j−1,n+1

)x j−1,n+1 − n j

)(x1,n+1 − x2,n+1

)There are two cases with respect to value of x1,4 as the following. In case ofx1,n+1 > x2,n+1, thecondition is:

F2,n+1 < mi−1

On the other hand, the condition evolves into the following expression in case of x1,n+1 < x2,n+1.

F2,n+1 > mi−1

Note that, the decision variable is x1,n+1 at t = 1. Therefore the first derivative of[yi]− with respect

to x1,n+1 is to be analyzed. Moreover, F1,n+1 is known, while F j,n+1 is a random stochastic variable att = 1 while j = 2, 3, 4, ..., i − 2, i; as a result recursive integral will be used to calculate the expectedfinancial distress cost.

• Suppose that x1,n+1 > x2,n+1:

First derivative of r[yi]− when x1,n+1 > x2,n+1:

d(EφF

([yi]−))

dx1,n+1=

d∫ +∞

Fi,n+1=0 ...∫ +∞

F3,n+1=0

∫ mi−1F2,n+1=0 (−yi) φ

QF2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1 ...φ

QFi,n+1

(Fi,n+1

)dFi,n+1

dx1,n+1

=

+∞∫Fi,n+1=0

...

+∞∫F3,n+1=0

mi−1∫F2,n+1=0

(F1,n+1 − F2,n+1

)φ

QF2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1 ...φ

QFi,n+1

(Fi,n+1

)dFi,n+1

• Suppose that x1,n+1 < x2,n+1:

First derivative of r[yi]− when xn+1 < xn+1:

d(EφF

([yi]−))

dx1,n+1=

d∫ +∞

Fi,n+1=0 ...∫ +∞

F3,n+1=0

∫ +∞

F2,n+1=mi−1(−yi) φ

QF2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1 ...φ

QFi,n+1

(Fi,n+1

)dFi,n+1

dx1,n+1

=

+∞∫Fi,n+1=0

...

+∞∫F3,n+1=0

+∞∫F2,n+1=mi−1

(F1,n+1 − F2,n+1

)φ

QF2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1 ...φ

QFi,n+1

(Fi,n+1

)dFi,n+1

71

Time n+1 is the time for delivery of the assets in the agreement of the firm. After all those observations,a generalized examination is necessary to produce a general formula for this instance. The value of theterm yn+1 is defined as the following:

yn+1 = yn +(S n+1 − Fn,n+1

)x3,n+1 +

(p − S n+1 − λ

)ξ − nn+1

= y1 +(F2,n+1 − F1,n+1

)x1,4 +

(F3,n+1 − F2,n+1

)x2,n+1 − n2 − n3 +

(S n+1 − Fn,n+1

)xn,n+1

+(p − S n+1 − λ

)ξ − nn+1 +

∑nj=4

((F j,n+1 − F j−1,n+1

)x j−1,n+1 − n j

)Negativity condition of yn+1 should be found by assuming that yn+1 < 0. By using Lemma 2, it isalready known that: x∗3,4 = ξ. Furthermore, Theorem 2 suggests that

(x2,4 − ξ

)5 0 regardless of the

value of the other variables.

There are two different conditions for the different cases with respect to the value of x∗1,4. For the sakeof ease of notation ω is defined such asω = −

∑nj=4

((F j,n+1 − F j−1,n+1

)x j−1,n+1 − n j

). Then, the variable breakpoint value mn will be defined

such that:

mn =Γ + ω −

∑nj=4

((F j,n+1 − F j−1,n+1

)x j−1,n+1 − n j

)(x1,n+1 − x2,n+1

)Finally, the negativity condition for x1,n+1 > x2,n+1 is shown below.

F2,n+1 < mn

The negativity condition for x1,n+1 < x2,n+1 is shown below.

F2,n+1 > mn

• Suppose that x1,n+1 > x2,n+1:

The first derivative of[y4

]− with respect to x1,4 is shown below.

d(EφF

([yn+1

]−))dx1,n+1

=

d∫ +∞

Fn,n+1=0 ...∫ +∞

F3,n+1=0

∫ mnF2,n+1=0 (−yn+1) φQ

F2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1 ...φ

QFi,n+1

(Fn,n+1

)dFi,n+1

dx1,n+1

=

+∞∫Fn,n+1=0

...

+∞∫F3,n+1=0

mn∫F2,n+1=0

(F1,n+1 − F2,n+1

)φ

QF2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1 ...φ

QFi,n+1

(Fn,n+1

)dFi,n+1

• Suppose that x1,n+1 < x2,n+1:

The first derivative of[y4

]− with respect to x1,4 is shown below.

d(EφF

([yn+1

]−))dx1,n+1

=

d∫ +∞

Fn,n+1=0 ...∫ +∞

F3,n+1=0

∫ +∞

F2,n+1=mn(−yn+1) φQ

F2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1 ...φ

QFi,n+1

(Fn,n+1

)dFi,n+1

dx1,n+1

=

+∞∫Fn,n+1=0

...

+∞∫F3,n+1=0

+∞∫F2,n+1=mn

(F1,n+1 − F2,n+1

)φ

QF2,n+1

(F2,n+1

)dF2,n+1φ

QF3,n+1

(F3,n+1

)dF3,n+1 ...φ

QFi,n+1

(Fn,n+1

)dFi,n+1

72

The convexity of the function may be proved by using the same methodology used in Theorem 2.Similar to the previous models, it is observed that the expressions

[y2

]− ,[y3]−,...,

[yn

]−,and[yn

]−arepiecewise convex functions and other variables in −4V2 (y1) are all linear. It is trivial that if a func-tion is convex, its expected value function is also convex. Consequently, it is obvious that the term−4V2 (y1) is convex in this model.

After a similar examination to the Theorem 9, it is observed that there are 4 different cases with respectto the decision variable.

• Case I: x1,n+1 > x2,n+1 and x1,n+1 > 0: In this case, the value of d(−4V2(y1))dx1,n+1

is observed to bepositive. It means that the expected total financial distress cost will increase while x1,n+1 isincreasing. Indeed, an extreme point cannot exist here, as a result it is not the optimum interval.

• Case II: x1,n+1 < x2,n+1 and x1,n+1 > 0: The expressiond

dx1,n+1(−4V2 (y1)) can either be positive or negative or zero. Therefore, the optimum decision

may exist in this interval.• Case III: x1,n+1 > x2,n+1 and x1,n+1 < 0: This case suggests that x2,n+1 < 0. However, this

assumption cannot be a valid one because it is already known that x2,n+1 ≥ 0 by using Theorem9. As a result, there is no need to examine this situation.

• Case IV: x1,n+1 < x2,n+1 and x1,n+1 < 0: In this case, the value of d(−4V2(y1))dx1,n+1

can only be nega-tive and it means that the total expected financial distress cost always increases while x1,n+1 isincreasing in the interval (−∞, 0]. Consequently, there cannot be an optimum decision in thisinterval and optimum x∗1,n+1 should be equal to or higher than 0 because of the convexity.

After examining all the possible cases with respect to the expected values of the decision variables,it is observed that the optimum decision policy is underhedging, such that 0 5 x∗1,n 5 x∗2,n 5 ... 5x∗n−2,n 5 x∗n−1,n = ξ.

B.11 Theorem 11

It is already stated that Case 2 leads to the optimum solution. When 4 approaches to ±∞, upperand lower limits of the integrals goes to minus or positive infinity too. Therefore, the first derivativeof −4V2 (y1) with respect to x1,4 is given by the equation below and it is equal to 0 because of thecharacteristics of normal distribution.

ddx1,4

(−4V2 (y1)) = lim4→±∞

e∫F2,4=0

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4

+ lim4→±∞

+∞∫F3,4=0

+∞∫F2,4=

g−F3,4 x2,4f

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

+ lim4→±∞

+∞∫F3,4=0

+∞∫F2,4=

k−F3,4 lf

(F1,4 − F2,4

)φQ

F2,4

(F2,4

)dF2,4φ

QF3,4

(F3,4

)dF3,4

= 0

73

B.12 Theorem 12

In order to prove the theorem, firstly the behavior of the term of −4V2 (y1) should be examined, similarto the proof of Theorem 2. The term −4V2 (y1) at t = 1 can be described mathematically as thefollowing after replacing the term x2,3 by ξ due to Lemma 2 ( the optimum decision quantity for one-period model):

−4V2 (y1) =

∞∫0

[n2 −

(F2,3 − F1,3

)x1,3 + r

[y2

]−y + β

∞∫0

[n3 −

(p − S 3 − λ

)ξ −

(S 3 − F2,3

)ξ + r

[y3

]−] φQS 3

(S 3) dS 3

]φQ

F2,3

(F2,3

)dF2,3

wherey2 = y1 − n2 +

(F2,3 − F1,3

)x1,3

y3 = y2 − n3 +(p − S 3 − λ

)ξ +

(S 3 − F2,3

)ξ − r

[y2

]−In order to find the optimum hedging quantity, it is necessary to analyze the first and the secondderivatives of the term −4V2 (y1) with respect to the decision variable at t = 1, i.e. x1,3. Since F1,3 =∫ ∞

0

[(F2,3

)]φQ

F2,3dF2,3 and the rest of the expression does not contain x1,3 except financial distress cost

premiums, the first derivative of the term −4V2 (y1) is:

d (−4V2 (y1))dx1,3

=

d∫ ∞

0

[r[y2

]−+

∫ ∞0 βr

[y3

]− φQS 3

(S 3) dS 3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

In order to examine the expression above, the breakpoints where y2 and y3 become zero should befound. Same notations used in the simple interest case are used also for breakpoints in this case. Thecondition that makes the expression y2 = y1 − n2 +

(F2,3 − F1,3

)x1,3 less than zero is the following

where b =−y1+F1,3 x1,3+n2

x1,3.

y2 < 0⇔

F2,3 < b i f x1,3 > 0

F2,3 > b i f x1,3 < 0

Since the expected financial distress cost at the end of the second period is also aimed to be found, theconditions that satisfy y3 < 0 should be examined with respect to the negativity condition of y2. In thecase of y2 > 0, y3 will be equal to y1 +

(F2,3 − F1,3

)x1,3 +

(p − F2,3 − λ

)ξ − n3 − n2. The condition of

y3 < 0 where y2 > 0 and c =−y1+F1,3 x1,3−(p−λ)ξ+n2+n3

x1,3−ξis:.

y3 < 0⇔

F2,3 < c i f x1,3 > ξ and y2 > 0

F2,3 > c i f x1,3 < ξ and y2 > 0

74

Similarly, y3 becomes equal to(y1 +

(F2,3 − F1,3

)x1,3 − n2

)(1 + r) +

(S 3 − F2,3

)ξ

+(p − S 3 − λ

)ξ − n3 when y2 < 0. Thus the condition of y3 < 0 where y2 < 0 and

d =(−y1+F1,3 x1,3+n2)(1+r)−(p−λ)ξ+n3

x1,3(1+r)−ξ is:

y3 < 0⇔

F2,3 < d i f x1,3 > ξ/ (1 + r) and y2 < 0

F2,3 > d i f x1,3 < ξ/ (1 + r) and y2 < 0

We will prove that the firm should underhedge by contradiction, by supposing x1,3 > ξ. The value ofd will be composed with respect to the values of b and c. Besides, the only intervals where variablesb and c are larger than zero will be examined because hedging will become a value neutral action inother case (the fact denoted by Theorem 7 and 8 is also valid for this model).

Decision Interval I: x1,3 > ξ

Case I: b > c

In this case, we have c < d < b. As a result, the expression will be as the following.

d (−4V2 (y1))dx1,3

=dr

∫ b

0

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ d

0

[(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)(1 + r) −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

By applying Leibniz Rule, the following expression is obtained.

ddx1,3

(−4V2 (y1)) = r

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 + βr

d∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

The expression above is definitely positive. As a result, decreasing hedging amount will also decreaseexpected financial distress cost.

Case II: b < c

In this case, we have b < d < c. As a result, the expression will be as the following.

d (−4V2 (y1))dx1,3

=dr

∫ b

0

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ b

0

[(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)(1 + r) −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ c

b

[−y1 +

(F1,3 − F2,3

)x1,3 + n2 −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

This expression can be transformed into:

75

d (−4V2 (y1))dx1,3

=dr

∫ b0

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ c0

[−y1 +

(F1,3 − F2,3

)x1,3 + n2 −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr2

∫ b0

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

After applying the Leibniz Rule, we obtain:

d (−4V2 (y1))dx1,3

= r

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

c∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr2

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

The expression above is also positive. Consequently, all of the first derivatives are positive in bothcases if b, c , d are larger than zero. Then, hedging amount should be decreased as long as x1,3 > ξ.

Decision Interval II: x1,3 < 0

This case means that the company gets short position. Again, there are 2 different cases with respectto values of b and c.

Case I: b > c

In this case, we have c < d < b. As a result, the expression will be as the following.

d (−4V2 (y1))dx1,3

=dr

∫ ∞b

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ ∞b

[(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)(1 + r) −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ b

c

[(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)−

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

This expression can be transformed into:

76

d (−4V2 (y1))dx1,3

=dr

∫ ∞b

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr2

∫ ∞b

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ ∞c

[(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)−

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

By applying Leibniz Rule, we obtain:

d (−4V2 (y1))dx1,3

= r

∞∫b

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

∞∫c

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr2

∞∫b

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

The expression above is definitely negative, therefore the decision variable should be increased in thiscase.

Case II: b < c

In this case, we have got b < d < c. As a result, the expression will be as the following.

d (−4V2 (y1))dx1,3

=dr

∫ ∞b

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ ∞d

[(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)(1 + r) −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

After applying the Leibniz Rule, we get:

d (−4V2 (y1))dx1,3

= r

∞∫b

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 + βr

∞∫d

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

The expression above is also negative. Consequently, all of the first derivatives are negative in bothcases if b, c , d are larger than zero and x1,3 < 0. Thus we should increase x1,3 if x1,3 < 0

Now let us examine the convexity of −4V2 (y1). Similar to the model with simple interest rates, all theterms in −4V2 (y1) except financial distress cost are linear. It is already known that the terms related tofinancial distress costs are piecewise convex, it means that all terms in the function are convex. Since

77

if a function is convex, its expected function is also convex; t −4V2 (y1) is concluded to be convex inx1,3.

In conclusion, we increase x1,3 in the region x1,3 < 0 and we decrease x1,3 when x1,3 > ξ; and thefunction is convex in x1,3. Therefore, the optimal solution must be in the region 0 5 x1,3 5 ξ.

B.13 Theorem 13

In the case that 4 = y1 − n2 = 0, the variables are realized as b = F1,3, c =(p−λ)ξ−F1,3 x1,3−n3

ξ−x1,3and d =

(p−λ)ξ−F1,3 x1,3(1+r)−n3

ξ−x1,3(1+r) . Now suppose that x1,3 = 0, then the variables are: b = F1,3 and c = d =(p−λ)ξ−n3

ξ.

Consequently, the first derivative of the term −4V2 (y1) is:

ddx1,3

(−4V2 (y1)) = βr( F1,3∫

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 +

∞∫(p−λ)ξ−n3

ξ

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

)

Similar to Theorem 4, it is known that this term is positive. Thus, the financial distress costs increasewhile x1,3 increases when x1,3 = 0 which proves that the company should not hedge at all, i.e. x∗1,3 = 0when 4 = 0.

B.14 Theorem 14

For this theorem, we will prove that there cannot be an optimal solution such that ξ/ (1 + r) < x1,3 < ξ

when 4 < 0.

Decision Interval III: ξ/ (1 + r) < x1,3 < ξ

Case I: b > c

When b > c, both numerator and denominator of c are negative while it is vice versa for b. Therefore,it is also obvious that the denominator and numerator of d are positive, and d > b. As a result, the firstderivative will be as the following.

ddx1,3

(−4V2 (y1)) =dr

∫ b

0

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ b

0

[(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)(1 + r) −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ ∞b

[−y1 +

(F1,3 − F2,3

)x1,3 + n2 −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

Then the expression can be transformed into the following by Leibniz Rule:

78

ddx1,3

(−4V2 (y1)) = r (1 + βr)

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 + βr

∞∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

= r (1 + βr)

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

Since the first derivative is always positive in the region, there cannot be an optimum solution here.

Case II: b < c

When b < c, both numerator and denominator of c are negative while it is vice versa for b. Therefore,the denominator of d is positive while the numerator may either be positive or negative. Thus, d < b.

As a result, the first derivative of −4V2 (y1) is:

ddx1,3

(−4V2 (y1)) =dr

∫ b0

(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ max(0,d)0

[(−y1 +

(F1,3 − F2,3

)x1,3 + n2

)(1 + r) −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

+dβr

∫ ∞c

[−y1 +

(F1,3 − F2,3

)x1,3 + n2 −

(p − F2,3 − λ

)ξ + n3

]φQ

F2,3

(F2,3

)dF2,3

dx1,3

The expression above can be simplified into the following by using Leibniz Rule:

ddx1,3

(−4V2 (y1)) = r

b∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 + βr

∞∫c

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr

max(0,d)∫0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

b is equal to F1,3 − 4/x1,3 which is larger than F1,3, since 4 < 0. Thus, F1,3 < b < c. Notethat, the biggest value of the expression

∫ i0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 realizes at i = F1,3. While

drifting away from F1,3, the value of the expression decreases, and finally it reaches to zero in bothends. Due to F1,3 < b < c and β < 1, the following the expression r

∫ b0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3

+βr∫ ∞

c

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 becomes larger than zero. In addition,

βr∫ max(0,d)

0

(F1,3 − F2,3

)φQ

F2,3

(F2,3

)dF2,3 is also a nonnegative variable. Consequently, the whole ex-

pression is concluded to be positive when 4 < 0 and there cannot be an optimal solution in the regionof ξ/ (1 + r) < x1,3 < ξ. On the other hand, the expression may be equal to zero when 4 > 0.

In conclusion, the first derivative never reaches to zero and there cannot exist an optimum decisionsuch as ξ/ (1 + r) < x∗1,3 < ξ when 4 < 0. Yet such an optimal decision is possible when 4 > 0.

79

B.15 Theorem 15

The objective function of the problem can be written more implicitly as:

V2 (y1) = minx1,2

∞∫0

[−4V2 (y1)

]φQ

S 2(S 2) dS 2

= minx1,2

[ ∞∫F1,2

(n2 −

(p − S 2 − λ

)ξ + hx1,2 −

[S 2 − F1,2

]+x1,2 + r

[y′

1

]−+ βr

[y2

]−) φQS 2

(S 2) dS 2

+

F1,2∫0

(n2 −

(p − S 2 − λ

)ξ + hx1,2 + r

[y′

1

]−+ βr

[y2

]−) φQS 2

(S 2) dS 2

]

Firstly, the conditions to face financial distress costs should be examined. We will prove the theoremby contradiction. The condition for incurring a financial distress cost due to the negativity of y

′

1 is:

y1

h< x1,2

Additionally, financial distress cost incurs due to y2 when:

y1 − n2 +(p − S 2 − λ

)ξ − hx1,2 +

[S 2 − F1,2

]+x1,2 < 0

However, two different cases occur with respect to the value of S 2. When the realized derivative priceis higher than the strike price of the underlying derivative, i.e. S 2 > F1,2, and x1,2 > ξ, the negativitycondition of y2 is the following:

S 2 <−y1 + n2 − (p − λ) ξ + hx1,2 + F1,2x1,2(

x1,2 − ξ)

On the other hand, the condition is the following for S 2 > F1,2 and decision x1,2 < ξ:

S 2 >−y1 + n2 − (p − λ) ξ + hx1,2 + F1,2x1,2(

x1,2 − ξ)

The term −y1+n2−(p−λ)ξ+hx1,2+F1,2 x1,2

(x1,2−ξ) may be written as F1,2 +−y1+n2−(p−λ−F1,2)ξ+hx1,2

(x1,2−ξ) to benefit from the

characteristics of normal distribution. Additionally, −y1+n2−(p−λ)ξ+hx1,2+F1,2 x1,2

(x1,2−ξ) will be denoted as a, and−y1+n2−(p−λ−F1,2)ξ+hx1,2

(x1,2−ξ) will be denoted as θ1 for ease in illustration.

For S 2 < F1,2, the financial distress cost occurs at the following condition.

S 2 >−y1 + n2 − (p − λ) ξ + hx1,2

−ξ

80

Since −y1+n2−(p−λ)ξ+hx1,2

−ξmay be written as F1,2 +

−y1+n2−(p−λ−F1,2)ξ+hx1,2

−ξ, −y1+n2−(p−λ)ξ+hx1,2

−ξmay be called

as b and −y1+n2−(p−λ−F1,2)ξ+hx1,2

−ξmay be called as θ2.

In order to prove the theorem, x1,2 is divided into two pieces. Suppose that x1,2 = xu1,2 + xo

1,2 such thaty1h = xu

1,2; and xo1,2 = 0 when y1

h = xu1,2. Here, xu

1,2 denotes the magnitude of call options under thebudget. On the other hand xo

1,2 represents call options bought over the budget. As stated, the theoremwill be proved by contradiction by supposing xo

1,2 > 0. Thus, the first derivative of the objectivefunction should be examined in two different intervals with respect to xo

1,2. If the result is positive, thefirm should not hedge more than y1

h units.

In case that(x1,2 > ξ

)and y1

h < x1,2, the value of −4V2 (y1) is the following:

−4V2 (y1) =

∞∫max(a,F1,2)

(n2 −

(p − S 2 − λ

)ξ + hx1,2 −

(S 2 − F1,2

)x1,2 − ry

′

1

)φQ

S 2(S 2) dS 2

+

max(a,F1,2)∫F1,2

(n2 −

(p − S 2 − λ

)ξ + hx1,2 −

(S 2 − F1,2

)x1,2 − ry

′

1 − βry2

)φQ

S 2(S 2) dS 2

+

F1,2∫min(b,F1,2)

(n2 −

(p − S 2 − λ

)ξ + hx1,2 − ry

′

1 − βry2

)φQ

S 2dS 2

+

min(b,F1,2)∫0

(n2 −

(p − S 2 − λ

)ξ + hx1,2 − ry

′

1

)φQ

S 2dS 2

=(n2 −

(p − S 2 − λ

)ξ)− βry

′

1 −

max(a,F1,2)∫F1,2

βry2φQS 2

dS 2 −

F1,2∫min(b,F1,2)

βry2φQS 2

(S 2) dS 2

Now, the first derivative of the expression above with respect to xo1,2 should be checked in order to

understand whether hedging over the budget is profitable or not.

d (−4V2 (y1))dxo

1,2=

d[(

n2 −(p − S 2 − λ

)ξ)− βry

′

1

]dxo

1,2

−

∫ max(a,F1,2)

F1,2βry2φ

QS 2

dS 2 +∫ F1,2

min(b,F1,2) βry2φQS 2

(S 2) dS 2

dxo1,2

=−βr d

[∫ ∞0

(−hxo

1,2

)φQ

S 2(S 2) dS 2

]dxo

1,2

+

−βr d[∫ max(a,F1,2)

F1,2

(y1 − n2 +

(p − S 2 − λ

)ξ − hx1,2 +

(S 2 − F1,2

)x1,2

)φQ

S 2(S 2) dS 2

]dxo

1,2

+

−βr d[∫ F1,2

min(b,F1,2)(y1 − n2 +

(p − S 2 − λ

)ξ − hx1,2

)φQ

S 2(S 2) dS 2

]dxo

1,2

= β

rh +

max(a,F1,2)∫F1,2

(h −

(S 2 − F1,2

))φQ

S 2dS 2 +

F1,2∫min(b,F1,2)

hφQS 2

dS 2

81

= βr[ ∞∫F1,2

(S 2 − F1,2

)φQ

S 2(S 2) dS 2

max(a,F1,2)∫

F1,2

φQS 2

dS 2 +

F1,2∫min(b,F1,2)

φQS 2

(S 2) dS 2 + 1

−

∞∫max(a,F1,2)

(S 2 − F1,2

)φQ

S 2(S 2) dS 2

]

Here, the result is always positive regardless of the values of a and b, due to the characteristics ofnormal distribution. As a result, the company should decrease the value of xo

1,2 to its minimum value,zero when x1,2 > ξ.

Then the second decision, under-hedging, should be analyzed. The value of −4V2 (y1) will be as thefollowing when x1,2 < ξ.

−4V2 (y1) =

max(a,F1,2)∫F1,2

(n2 −

(p − S 2 − λ

)ξ + hx1,2 −

[S 2 − F1,2

]+x1,2 − ry

′

1

)φQ

S 2(S 2) dS 2

+

∞∫max(a,F1,2)

(n2 −

(p − S 2 − λ

)ξ + hx1,2 −

[S 2 − F1,2

]+x1,2 − ry

′

1 − βry2

)φQ

S 2(S 2) dS 2

+

F1,2∫min(b,F1,2)

(n2 −

(p − S 2 − λ

)ξ + hx1,2 − ry

′

1 − βry2

)φQ

S 2(S 2) dS 2

+

min(b,F1,2)∫0

(n2 −

(p − S 2 − λ

)ξ + hx1,2 − ry

′

1

)φQ

S 2(S 2) dS 2

=(n2 −

(p − S 2 − λ

)ξ)−

∞∫0

βry′

1φQS 2

(S 2) dS 2

−

∞∫max(a,F1,2)

βry2φQS 2

(S 2) dS 2 −

F1,2∫min(b,F1,2)

βry2φQS 2

(S 2) dS 2

Let us take the first derivative of the expression with respect to xo1,2

d (−4V2 (y1))dxo

1,2=

d[(

n2 −(p − S 2 − λ

)ξ)−

∫ ∞0βry

′

1φQS 2

dS 2

]dxo

1,2

−

∫ ∞

max(a,F1,2) βry2φQS 2

dS 2 +∫ F1,2

min(b,F1,2) βry2φQS 2

(S 2) dS 2

dxo1,2

=−βr d

[∫ ∞0

(−hxo

1,2

)φQ

S 2(S 2) dS 2

]dxo

1,2

+

−βr d[∫ ∞

max(a,F1,2) β(y1 − n2 +

(p − S 2 − λ

)ξ − hx1,2 +

(S 2 − F1,2

)x1,2

)φQ

S 2(S 2) dS 2

]dxo

1,2

82

+

−βr d[∫ F1,2

min(b,F1,2)(y1 − n2 +

(p − S 2 − λ

)ξ − hx1,2

)φQ

S 2(S 2) dS 2

]dxo

1,2

= βr

∞∫F1,2

(S 2 − F1,2

)φQ

S 2(S 2) dS 2

∞∫

max(a,F1,2)

βφQS 2

(S 2) dS 2 +

F1,2∫min(b,F1,2)

φQS 2

(S 2) dS 2 + 1

−

max(a,F1,2)∫F1,2

(S 2 − F1,2

)φQ

S 2(S 2) dS 2

Here, the result is always positive regardless of the values of a and b. Therefore, the company shoulddecrease the value of xo

1,2 to its minimum value, zero. Since both cases suggest the same result, it canbe said that the company should not have a decision such as y1

h < x1,2.

B.16 Theorem 16

At the proof of the Theorem 15, the terms a, b, θ1 and θ2 are defined. It is already known that both ofthe terms θ1 and θ2 have the same expression −y1 + n2 −

(p − λ − F1,2

)ξ + hx1,2 as numerator. As a

result, both of the terms θ1 and θ2 have to be in the opposite sign when x1,2 > ξ. Thus there are threecases that can be valid when x1,2 < ξ as the following.

• a > F1,2, b < F1,2

• a < F1,2, b > F1,2

• a = F1,2, b = F1,2

Similarly, the terms θ1 and θ2 have to be in the same sign when x1,2 < ξ. Thus there are three casesthat can be valid when x1,2 < ξ.

• a > F1,2, b > F1,2

• a < F1,2, b < F1,2

• a = F1,2, b = F1,2

We also know that the term −4V2 (y1) is convex with respect to x1,2. The value of −4V2 (y1) is shownbelow.

−4V2 (y1) =(n2 −

(p − S 2 − λ

)ξ)

+

∞∫0

(hx1,2 −

[S 2 − F1,2

]+x1,2 + βr

[y2

]−) φQS 2

(S 2) dS 2

Since the term(n2 −

(p − S 2 − λ

)ξ)

is linear, which means it is both convex and concave. When the

probability distribution functions are ignored, it is observed that the expression hx1,2−[S 2 − F1,2

]+x1,2+

βr[y2

]− is also convex. It is known that if a function is convex, its expected function is also convex; asa result −4V2 (y1) is concluded to be convex.

Since −4V2 (y1) is convex and optimum solutions may exist in different intervals of x1,2, we willprovide just one optimum region that satisfy optimality conditions in some realistic cases.

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After all these analysis, two different decision should be examined by getting the first derivative of theobjective function with respect to x1,2.The examination of the decision x1,2 > ξ

Decision I: x1,2 = ξ

−4V2 (y1) =

∞∫max(a,F1,2)

(n2 −

(p − S 2 − λ

)ξ + hx1,2 −

(S 2 − F1,2

)x1,2

)φQ

S 2(S 2) dS 2

+

max(a,F1,2)∫F1,2

(n2 −

(p − S 2 − λ

)ξ + hx1,2 −

(S 2 − F1,2

)x1,2 − βry2

)φQ

S 2(S 2) dS 2

+

F1,2∫min(b,F1,2)

(n2 −

(p − S 2 − λ

)ξ + hx1,2 − βry2

)φQ

S 2(S 2) dS 2

+

min(b,F1,2)∫0

(n2 −

(p − S 2 − λ

)ξ + hx1,2

)φQ

S 2(S 2) dS 2

=(n2 −

(p − S 2 − λ

)ξ)−

max(a,F1,2)∫F1,2

βry2φQS 2

(S 2) dS 2 −

F1,2∫min(b,F1,2)

βry2φQS 2

(S 2) dS 2

The value of the expression above changes with respect to the values of a and b. Therefore, subcasesshould be analyzed separately.

Decision I.II: a < F1,2, b > F1,2 and x1,2 = ξ

First derivative:

d (−4V2 (y1))dx1,2

=

d[(

n2 −(p − S 2 − λ

)ξ)−

∫ max(a,F1,2)F1,2

βry2φQS 2

(S 2) dS 2

]dx1,2

−

d(∫ F1,2

min(b,F1,2) βry2φQS 2

(S 2) dS 2

)dx1,2

=

−βr d[∫ ∞

F1,2(y1 − n2 + (p − S 2 − λ) ξ) φQ

S 2(S 2) dS 2

]dx1,2

= 0

The cluster of the decisions that satisfies the condition above may cause multi-optimality because allare local extreme points. If the problem is proved to be convex, it means that this is the optimumdecision.

Decision I.III: a = F1,2, b = F1,2 and x1,2 = ξ

It is the same with case a < F1,2, b > F1,2. It is always zero, as a result there is an extreme point here.

84

Now, it is analyzed that both of the cases always provide an optimal solution. It is already knownthat the conditions satisfying a 5 F1,2, b = F1,2 should also satisfy θ1 5 0, θ2 = 0. Therefore, theexpression −y1 + n2 −

(p − λ − F1,2

)ξ + hx1,2 has to be less than zero because it is the numerator of

both θ1 and θ2 terms. Then, another condition may be written for optimal hedging decision at time 1such as x1,2 <

y1h +

(p−λ−F1,2)ξ−n2

h .

It is observed that the decision of x∗1,2 = ξ is an optimal decision as long as x∗

1,2 <y1h +

(p−λ−F1,2)ξ−n2

h ;and also the budget constraint at time 1, which is x∗1,2 < y1

h , are satisfied together. Two maximum

limit conditions may be united as x∗1,2 5 y1h +

((p−λ−F1,2)ξ−n2

h

)−. Therefore, the optimum decision is the

decisions satisfying y1h +

((p−λ−F1,2)ξ−n2

h

)−= x∗1,2 = ξ as long as y1

h +

((p−λ−F1,2)ξ−n2

h

)−is more than ξ.

85